Transcript Slide 1

The Game of Algebra
or
The Other Side of
Arithmetic
Lesson 20
by
Herbert I. Gross & Richard A. Medeiros
© 2007 Herbert I. Gross
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More
on
Inverse
Functions
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© 2007 Herbert I. Gross
The concept of functions, including inverse
functions and composition of functions,
gives us an effective way to explain the
informal process of “undoing” in a more
mathematically precise way.
For example, we have viewed subtraction as
being “unaddition”. The more formal view
is to refer to subtraction as being the
inverse of addition. There is a strong
connection between the word “inverse” as it
is used here and the word “inverse” as it is
used in the term “inverse function”. next
© 2007 Herbert I. Gross
For example, to “undo” adding 3 we
subtract 3. That is: if we add 3 to a given
number and then subtract 3 the result is the
given number. In less formal terms, “we are
back to where we started from”.
In the language of functions, the instruction
“Add 3” can be represented by the function f
where f(x) = x + 3. The instruction
“Subtract 3” can be represented by the
function g where g(x) = x – 3.
So if our input is 7, we see that f(7) = 10, and
g(f(7)) = g(10) = 7. In a similar way, we see
that g(7) = 4 and, f(g(7)) = f(4) = 7.
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© 2007 Herbert I. Gross
There was nothing special about our choice
of 7 as our input. That is: with f and g
defined by f(x) = x + 3, and g(x) by
g(x) = x – 3; for any number x,
f(g(x)) = g(f(x)) = x. More specifically…
f(g(x)) = f (g(x))
(x – 3) = (x – 3) + 3 = x
and…
g(f(x)) = g (x
(f(x))
+ 3) = (x + 3) – 3 = x
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© 2007 Herbert I. Gross
The composition of these two functions, g
and f, is itself a function, and it is a
function that basically “does nothing”.
That is: if we let I denote fog (or in
this case, equivalently gof) we see that
I(x) = x. For this reason, we
refer to I as being the identity function.
© 2007 Herbert I. Gross
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Notes
Just as multiplying a number by 1 doesn’t
change the number, composing a function
with I doesn’t change the function.
That is, for any function, f …
x )
I( f(x) ) = ( f(x)) and f( I(x) ) = f( I(x)
Definition
If f(g(x)) = g(f(x)) = I(x) = x for all x in the
domain of f and the domain of g, we call
f and g inverses of one another and
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write f = g-1 and g = f-1.
© 2007 Herbert I. Gross
In essence, we obtain the inverse of a
function (if there is an inverse function) by
interchanging the input and the output (that
is, the domain and the image).
For example, suppose that…
f(x) = x + 3
If we interchange x and f(x) the equation
becomes…
x = f(x) + 3
and if we subtract 3 from both sides of our
equation we obtain…
f(x) = x – 3
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© 2007 Herbert I. Gross
The “tricky part” is that f(x) in the equation
f(x) = x – 3 is not the f(x) in the equation
f(x) = x + 3; rather it is its inverse because we
interchanged the roles of x and f(x). In other
words…
If f(x) = x + 3, then f-1(x) = x – 3
The confusion results from the tradition that
we usually denote the input of a function
by x. To avoid this confusion we may let
y = f(x) in the equation f(x) = x + 3 to obtain…
y = f(x) = x + 3
…and if we then solve for x in the above
equation we see that…
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x = y – 3 = g(y) = f-1(y)
© 2007 Herbert I. Gross
Not every function has an inverse.
For example, suppose we define a function
on the set of all 3-digit numbers such that
the output of the function is the sum of the
3 digits. Thus, for example,
f(123) = 1 + 2 + 3 = 6
However, there are several other 3-digit
numbers the sum of whose digits is 6.
They are… 105, 114, 123, 132, 141, 150, 204,
213, 222, 231, 240, 303, 312, 321, 330, 402,
411, 420, 501, 510 and 600. So for example,
123 ≠ 510, yet f(123) = f(510) = 6.
© 2007 Herbert I. Gross
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However, if the function had had an inverse, it
would mean that for each output there would
have been only one input that yielded that
output. In more mathematical terms, it means
that if a ≠ b, f(a) ≠ f(b).
Thus, in this case f-1 doesn’t exist.
If m ≠ 0, the linear function f, defined by
f(x) = mx + b always has an inverse. Namely, if
m ≠ 0, its graph is either always rising or
always falling. That is, if m > 0 the line is
always rising, and if m < 0 the line is
always falling.
In either case, this means that no two inputs
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can have the same output.
© 2007 Herbert I. Gross
To obtain the inverse of an “invertible”
function, we interchange the input and the
output. In terms of the graphs, we interchange
the points (x,y) and (y,x). In general, these two
points will be different.
y
For example,
the point (3,5)
is not the
same as the
point (5,3)
(3,5)
(5,3)
x
© 2007 Herbert I. Gross
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In fact, the only time that these two points
will actually be the same is when x = y.
Hence, the points of intersection of the
graphs of y = f(x) and y = f-1(x) will always
be on the line whose equation is y = x.
More specifically, the two graphs will be
symmetric with respect to the line y = x.
In other words, if we look at the graph
y = f(x) through a mirror that is placed on
the line y = x, the mirror image of this
graph will be the graph y = f-1(x).
© 2007 Herbert I. Gross
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This is illustrated below with the graphs
y = x + 3, y = x – 3, and y = x.
y=x+3
y=x
(6,9)
y=x–3
(3,6)
(9,6)
(0,3)
(6,3)
(-3,0)
(3,0)
(0,-3)
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For further practice, let’s draw the line…
y = 2x + 3
y = f(x) = 2x + 3,
…and reflect
it about the
line, y = x,
(3,9)
y=x
y = f-1(x)
…to obtain the
line, y = f-1(x),
(0,3)
(9,3)
Note that…
f-1(x) = x – 3
2
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(3,0)
(-3,-3)
Note
If m is negative, a similar analysis holds.
Namely if m is negative it means that as x
increases, f(x) decreases. Hence again,
no two different values for x can yield the
same value for f(x).
Geometrically, it means that any horizontal
line intersects the straight line that
represents the graph of f at one and only
one point.
© 2007 Herbert I. Gross
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Generalization
If a function is continuous (that is, if its
graph is “unbroken”) then it has an inverse
if and only if its graph is either
“always rising” or “always falling”.
If its graph is “always rising” we say that
the function is monotonically increasing;
and if its graph is “always falling” we say
that the function is
monotonically decreasing.
© 2007 Herbert I. Gross
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Key Point
In the language of functions we say that f is
monotonically increasing, if whenever a > b,
then f(a) > f(b)
In other words: as x increases f(x) also
increases ...
And we say that f is monotonically
decreasing if whenever a > b, then f(a) < f(b).
That is, as x increases f(x) decreases.
© 2007 Herbert I. Gross
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For example, with respect to the
function f where f(x) = 2x, notice that as
x increases so also does f(x). In other
words, the curve y = 2x which
represents the graph of f is always
rising. Hence, no horizontal line can
intersect the graph at more than
one point.
© 2007 Herbert I. Gross
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Key Point
In geometric terms, our above discussion
essentially means that if the curve that
represents the continuous function is
sometimes rising and sometimes falling,
then the function will not have an inverse.
Let’s illustrate this in the case for which
f is defined by f(x) = x2. Its graph is the
curve y = x2. Notice that while every
vertical line intersects this curve at one
and only one point, every horizontal line
that is above the x-axis intersects the
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curve in two places.
© 2007 Herbert I. Gross
Key Point
In other words, if this function had an
inverse, then for each output y there would
be one and only one value of x for which
(x,y) belongs to the graph.
Since f(x) = x2 and since there are two
values of x for which x2 = 4, it means that
f-1 doesn’t exist.
© 2007 Herbert I. Gross
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Namely, the graph of f is the curve y = x2.
y = x2
y
(-3,9)
(3,9)
(-2,4)
(2,4)
(0,0)
© 2007 Herbert I. Gross
x
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Notice that every vertical line intersects
this curve at one and only one point.
y
This is illustrated
for the line x = 2
which intersects
the curve only at
the point (2,4),
and the line x = -2
which intersects
the curve only at
the point (-2,4)
© 2007 Herbert I. Gross
(-2,4)
y = x2
(2,4)
x
x = -2
x=2
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On the other hand,
if instead, we start
with the horizontal
line y = 4, we see
that it intersects
the curve at two
different points,
namely (2,4) and
(-2,4).
© 2007 Herbert I. Gross
y
y = x2
y=4
(-2,4)
(2,4)
x
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Aside
At first glance it may seem that not all
vertical lines will intersect the curve y = x2.
Notice, however, that no matter how large
the number c is, the line y = c will intersect
the curve y = x2 at the point (c,c2)
© 2007 Herbert I. Gross
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Although the line y = 4 intersects the curve
in two points, notice that we can represent
y = x2 as the union of the two curves y = x2
where x is non-negative,
and y = x2 where x is negative.
y
(-3,9)
(3,9)
C2
C1
(-2,4)
(2,4)
(0,0)
© 2007 Herbert I. Gross
x
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The curves C1 and C2 are the
graphs of the functions g and h where
g(x) = x2 when x is non-negative and
h(x) = x2 when x is negative.
In other words, if g and h are defined as
above, g is a monotonically increasing
function and h is a monotonically decreasing
function.
Therefore, the inverses of both g and h exist.
© 2007 Herbert I. Gross
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In summary…
y = h(x)
y = g(x)
y
The line y = 4
intersects the
curve C1 only at
the point (2,4), and
intersects the
curve C2 only at
the point (-2,4).
© 2007 Herbert I. Gross
(-3,9)
(3,9)
C2
C1
y=4
(2,4)
(-2,4)
(0,0)
x
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Independently of whether the graph of a
function is a curve that is always rising or
always falling we may still interchange its
x- and y-coordinates. Whenever we do this,
the resulting curve will be the mirror image of
original curve with respect to the line
y = x, but it might not represent a function.
To make this idea more concrete let’s look
specifically at the curve x = y2 which represents
the mirror image of the curve y = x2 with
respect to the line y = x. In terms of functions,
this curve represents the function x = [f(x)]2.
© 2007 Herbert I. Gross
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Note
In our “plain English” version when we
define the function f by the equation
x = [f(x)]2 we are defining the function
implicitly.
That is: we are not explicitly telling what
the program is, but what we are saying is
that the program has the property that for
each input the square of the output is
always equal to the input.
© 2007 Herbert I. Gross
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Note
The concept of implicit and explicit is
not restricted to the study of functions.
For example, when we say that a
number is 4 less than 7, we have
implicitly defined the number 3.
© 2007 Herbert I. Gross
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The Curve x = y2
Suppose we want to draw the graph that
represents the equation…
x = j(y) = y2
Recalling that y2 cannot be negative, we see
from the above equation that for any value
of y, x must be at least as great as 0.
Hence, the graph exists only to the right
of the y-axis.
© 2007 Herbert I. Gross
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The Curve x = y2
To locate points
on the curve
x = y2, it’s
probably less
cumbersome to
pick values for y
and then to solve
for the
corresponding
value(s) of x.
© 2007 Herbert I. Gross
For example…
y
0
1
-1
2
-2
3
-3
4
-4
y2
0
1
1
4
4
9
9
16
16
x
0
1
1
4
4
9
9
16
16
(x,y)
(0,0)
(1,1)
(1,-1)
(4,2)
(4,-2)
(9,3)
(9,-3)
(16,4)
(16,-4)
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From our chart we see that the graph of
x = y2 is…
y
x = y2
(9,3)
(4,2)
x
(0,0)
(4,-2)
(9,-3)
© 2007 Herbert I. Gross
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The Curve x = y2
The fact that for each (positive) value of x
there are two corresponding values of y
means that x = y2 is not the graph of a
function.
More explicitly, if we take the square root of
both sides of the equation x = y2, we obtain
the equivalent equation…
y = ±√ x
© 2007 Herbert I. Gross
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“y = ±√ x ” is an abbreviation for…
“y = +√ x
or y = -√ x ”
In terms of a graph…
y
y = +√ x
(9,3)
(4,2)
x
(0,0)
(4,-2)
(9,-3)
© 2007 Herbert I. Gross
y = -√ x
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While the line x = 4 intersects the curve
x = y2 at both (4,2) and (4,-2); it intersects
“y = +√ x only at (4,2) and y = -√ x ”
only at (4,-2). y
x=4
(9,3)
(4,2)
yx == y+√2 x
x
(0,0)
(4,-2)
(9,-3)
© 2007 Herbert I. Gross
y = -√ x
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More generally, we see that the curve D1
defined by y = +√ x is always rising.
Hence, it represents a function which we
will denote by kp. That is: kp(x) = +√ x
In a similar way, we see that the curve D2
defined by y = -√ x is always falling.
Hence, it represents a function which we
will denote by kn. That is: kn(x) = -√ x
© 2007 Herbert I. Gross
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In Summary…
What we have shown thus far is that the
function f, defined by f(x) = x2 does not
possess an inverse.
However, it is the union of two functions, g
and h, both of which have an inverse.
More specifically…
g(x) = x2 when x is non-negative
and
h(x) = x2 when x is negative
© 2007 Herbert I. Gross
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In Summary…
g(x) = x2 when x is non-negative
h(x) = x2 when x is negative
g-1(x) = kp(x) = +√ x when x is non-negative
h-1(x) =kn(x) = -√ x when x is negative
© 2007 Herbert I. Gross
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In terms of a graph…
y=x
(-3,9)
y = h(x)
C2
(-2,4)
(0,0)
(4,-2)
(9,-3)
D2
y = h -1(x)
© 2007 Herbert I. Gross
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C1
y = g(x)
y=x
(3,9)
(-3,9)
y = h(x)
C2
y = h-1(x)
(2,4)
D1
(-2,4)
(9,3)
(4,2)
(0,0)
(4,-2)
(9,-3)
D2
y = h -1(x)
© 2007 Herbert I. Gross
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An Historical Note
Given a continuous curve drawn at random; it
is very unlikely that it is either always rising or
always falling.
Therefore, if a continuous curve is drawn at
random, it will probably not represent a
function that possesses an inverse.
For this reason, until relatively recently, it was
common to talk about multi-valued functions.
In that context if we let f(x) = x2, we would have
said that f-1 existed as a multi-valued function;
and the two functions kp(x) and kn(x) would
have been referred to as single valued
branches of f-1.
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© 2007 Herbert I. Gross
A Historical Note
Nowadays, however, we use the word
“function” to mean “single valued function”;
and we use the term “relation” rather than
“multi-valued function”. That is, we would
refer to f(x) = ± √x as being a relation.
The distinction between a relation and a multivalued function is not too important because
every relation (that is, a multi-valued function)
can be represented by the union of two or
more functions, each of which has an inverse.
This was illustrated in our above discussion
when we talked about trying to find the inverse
of f in the particular case where f(x) = x2.
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© 2007 Herbert I. Gross
An Enrichment Discussion
Suppose we had “invented”
Prelude
addition but had not yet
“invented” subtraction. In this case, we
would know that the function f, defined by
f(x) = x + 3 must have an inverse. Namely
because no two different values of x have
the same image, we see that once we know
the value of x + 3, we can uniquely
determine the corresponding value of x.
In terms of a graph, we could construct f-1
by first drawing the line y = x + 3 and then
reflecting it about the line y = x. The
resulting line would be the graph of f-1. next
© 2007 Herbert I. Gross
An Enrichment Discussion
Notice that since we can draw its graph, the
function f-1 exists even if we don’t bother to
give the function a specific name.
However, as we all know, it turned out that
we “invented” the minus sign and defined
f-1 by f-1(x) = x – 3.
Thus, to solve an equation such as 3 + x = 12
using a calculator, we would translate this
indirect computation into a
direct computation by rewriting it in the
equivalent form 12 – 3 = x.
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© 2007 Herbert I. Gross
An Enrichment Discussion
Of course this discussion
might seem boring to us
because we’ve know for a
long time, even if not in
these exact words, that
subtraction is the inverse
of addition.
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© 2007 Herbert I. Gross
An Enrichment Discussion
The above discussion applies almost
verbatim to any function f, provided that f is
either monotonically increasing or
monotonically decreasing. In particular it
applies to the monotonically increasing
function f where f(x) = 2x.
However, we are most likely not nearly as
comfortable discussing exponential
growth and its inverse as we are when we
are discussing addition and its inverse.
So let’s construct f-1, when f(x) = 2x.
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© 2007 Herbert I. Gross
An Enrichment Discussion
By way of review, starting with f(x) = 2x we
can choose various values for x and
compute the ordered pairs (x,2x). In this
way we see that the points (-1,1/2), (0,1), (1,2),
(2,4), (3,8)... belong to the curve y = 2x
(which represents the graph of f). By
interchanging x and y we see that the points
(1/2, -1), (1,0), (2,1), (4,2), (8,3)…belong to the
curve that represents the graph of f-1.
Or looking only at the geometric graph, we
start with the curve y = 2x and reflect it,
point by point, about the line y = x. In this
way the curve that is thus generated
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represents the graph of f-1
© 2007 Herbert I. Gross
Pictorially…
Thus, f-1
exists
whether or
not we give it
an explicit
name.
It turns out
that the
name we
give to f-1 in
this case is
f-1(x) = log2x
f(x) = 2x
y=x
(3,8)
(2,4)
(1,2)
(0,1)
(-1,1/ )
2
(2,1)
(1,0)
(1,-1/2)
(8,3)
(4,2) f-1(x) = log2x
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© 2007 Herbert I. Gross
An Enrichment Discussion
There was nothing special about choosing
the base to be 2 and then showing that if
f(x) = 2x, then f-1(x) = log2x.
Thus, if we replace 2 by b, we obtain the
more general result…
if f(x) = 2
b x, then f-1( x ) = log b2 x
Because of place value notation, we often
choose b to be 10. In this case we omit the
subscript and simply write log x rather than
log10x.
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© 2007 Herbert I. Gross
An Enrichment Discussion
For example…
23 =8 means the same thing as log28 = 3
52 = 25 means the same thing as log525 = 2
4-2 = 1/16 means the same thing as log41/16 = -2
103 = 1,000 means the same thing as log 1,000 = 3
Or with a shift in emphasis…
log636 = 2 means the same thing as 62 = 36
log381 = 4 means the same thing as 34 = 81
log21/8 = -3 means the same thing as 2-3 = 1/8
log 10,000 = 4 means the same thing as 104 =10,000
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© 2007 Herbert I. Gross
An Enrichment Discussion
If we try to use the calculator to find the
value of x if, say, x = log315, we discover
that there is no log3 key. It would have
been nice if there were such a key because
we could then enter 15 and press the log3
key to obtain the value of x.
However, as we discussed in our previous
lesson on exponential functions (Lesson
19), most calculators have a “log x” key.
Reminder…
The relationship between 10x and log x is
that if f(x) = 10x, f-1(x) = log x.
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© 2007 Herbert I. Gross
Solving the Equation bx = c
Where b and c Are Positive Constants
0.8451
7
log x
Let’s use a calculator that has a
“log x” key to solve the equation
10x = 7, we enter 7 on the calculator
and then press the “log x” key.
In this way, we obtain the result that
to 4 decimal place accuracy, x = 0.8451.
In other words, 100.8451 = 7.
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© 2007 Herbert I. Gross
The major problem is…
What happens when the base is not 10?
For example, suppose we want to find the
value of x for which 7x = 28. Knowing that
7 =(10
100.8451,) we may replace the equation…
7x = 28
by the equivalent equation…
7x = 28
and by our rules for exponents, we can
replace the above equation by the
equivalent equation…
100.8451x = 28
© 2007 Herbert I. Gross
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Solving the Equation bx = c
100.8451x = 28
Since log 28 means the power to which 10
has to be raised to obtain 28 and since the
above equation tells us this power is
0.8451x
0.8451x, we see that
the above equation
may be rewritten as…
1.447158
log 28 =
2
Using our calculator we see that…
8
log x
© 2007 Herbert I. Gross
log 28 = 1.447158
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Solving the Equation bx = c
So replacing log 28 by 1.447158 in the
equation…
log 28 = 0.8451x
we obtain the equivalent equation…
1.447158 = 0.8451x
And from the equation above, we see that…
x = 1.447158 ÷ 0.8451 = 1.71259
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© 2007 Herbert I. Gross
Checking the Solution for the Equation bx = c
As a check we see that 71.71259 = 28.00957...
Since 1.447158 = log
log 77,
log 28
28 and 0.8451 = log
the derivation that took us from the
equation 7x = 28 to the equation…
x = 1.447158 ÷ 0.8451
can be rewritten as…
If 7x = 28, x = 1.447158 ÷ 0.8451
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© 2007 Herbert I. Gross
Solving the Equation bx = c
If we replace 7 by b
b… and 28 by
y,
y y…
we obtain the more general result…
If 7 x = 28, x = log 28 ÷ log 7
For further practice, let’s apply the above
formula with b = 8 and y = 40 to find the
value of x for which…
8x = 40
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© 2007 Herbert I. Gross
Solving the Equation bx = c
To Solve the Equation 8x = 40
In this case,
the formula x = log y ÷ log b tells us that…
x = log 40 ÷ log 8
Using the “log x” key we see that…
log 40 = 1.60206 and log 8 = 0.903090
Therefore…
X = 1.60206 ÷ 0.903090 = 1.7740
As a check we see that 81.7740 = 40.002
© 2007 Herbert I. Gross
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This concludes our present
discussion of inverse functions.
Further practice is left for the
Key Stone Illustrations
and
the Exercise Set for this Lesson.
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© 2007 Herbert I. Gross