Transcript Logarithmic Functions - St. Edward's University

Logarithmic Functions
• Recall that for a > 0, a  1, the exponential function f(x) = ax
is one-to-one. This means that the inverse function exists, and
we call it the logarithmic function with base a, written loga.
• We have:
y  log a x

x  ay.
• Example. log10 1000 = 3, log10 10 = 0.5, log10 0.01 = –2.
• The notation log x is shorthand for log10 x, and log x is called
the common logarithm.
• The notation ln x is shorthand for loge x, and ln x is called the
natural logarithm.
Graphs of 10x and log x

Note that the domain of log x is the set of positive x values.
Logarithmic Identities and Properties
• From the definition of the logarithm it follows that
loga a  1
loga 1  0.
• From the fact that the exponential and logarithmic functions
are inverse functions it follows that
a loga x  x
loga a x  x.
• Since loga x is one-to-one, it follows that
loga u  loga v  u  v.
• Since the graphs of loga and logb intersect only at x = 1,
loga u  logb u  u  1 or a  b.
Fundamental Properties of Logarithms
• The following three properties of logarithms can be proved by
using equivalent exponential forms.
loga xy  loga x  loga y
x
loga    loga x  loga y
 y
loga x n  n loga x, n a real number.
• Problems. log 2 + log 5 = ???, log 250 – log 25 = ???,
log 101/3 = ???
Write the expression as a single logarithm
1
2
loga ( x  1)  2 loga ( x  1) 
1
2
loga ( x  1)  loga ( x  1) 2 
x 1
loga
( x  1) 2
Solving an equation using logarithms
• If interest is compounded continuously, at what annual rate
will a principal of $100 triple in 20 years?
A  Pert
300  100e 20 r
3  e 20 r
ln 3  ln e 20 r
ln 3  20r
ln 3
r
 0.055or 5.5%
20
Change of Base Formula
• Sometimes it is necessary to convert a logarithm to base a to
a logarithm to base b. The following formula is used:
loga x
logb x 
, a  0, b  0, a  1, b  1.
loga b
• Compute log2 27 using common logs and your calculator.
log 27 1.43136
log2 27 

 4.7549
log 2 0.30103
• Check your answer:
24.7549  27
Exponential Equations
• When solving an exponential equation, consider taking
logarithms of both sides of the equation.
• Example. Solve 32x–1 = 17.
log 32 x 1  log17
(2 x  1) log 3  log17
log17
2x 1 
log 3
 log17 
  1.7895
x  0.51 
log 3 

Solving an Exponential Equation for Continuous Compounding
• Problem. A trust fund invests $8000 at an annual rate of 8%
compounded continuously. How long does it take for the
initial investment to grow to $12,000?
Solution. We must solve for t in the following equation.
8000e 0.08t  12000
12000
0.08 t
e

 1.5
8000
0.08t  ln 1.5
ln 1.5
t
 5.07 years.
0.08
Logarithmic Equations
• When solving a logarithmic equation, consider forming a
single logarithm on one side of the equation, and then
converting this equation to the equivalent exponential form.
• Be sure to check any "solutions" in the original equation since
some of them may be extraneous.
• Problem. Solve for x.
log2 x  log2 ( x  2)  3
log2 x( x  2)  3
x( x  2)  23 , equivalentexponential form
x 2  2x  8  0
( x  2)(x  4)  0
x  2 or x  4.
Summary of Exponential and Logarithmic Functions; We discussed
•
•
•
•
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Definition of logarithm as inverse of exponential
The common logarithm
The natural logarithm
Graphs of y = log x and y = ax
Domain of the logarithm
Fundamental properties of logarithms
 log of product
 log of quotient
n
 log of x
• Change of base formula
• Solving exponential equations
• Solving logarithmic equations