Transcript ELF.01.9 - Solving Logarithmic Equations

ELF.01.9 - Solving
Logarithmic Equations
MCB4U - Santowski
(A) Introduction to Logarithmic
Equations
Many measurement scales used for naturally
occurring events like earthquakes, sound
intensity, and acidity make use of logarithms
 Additionally, since we know how logarithms and
exponents are related, many applications
involving exponents can also be mathematically
analyzed using logarithms
 So in working with these types of problems, we
need to know how to solve logarithmic
equations

(B) Example

The magnitude, R, of an earthquake on the
Richter scale is given by the equation R =
log(a/T) + B, where a is the amplitude of the
vertical ground motion (measured in microns), T
is the period of the seismic wave (measured in
seconds) and B is a factor that accounts for the
weakening of the seismic waves. Find the
amplitude of the vertical ground motion for an
earthquake that measured 6.3 on the Richter
scale, and the period of the seismic wave was
1.6 seconds and B = 4.2
(B) Example












Givens:
R = log(a/T) + B
R = 6.3
T = 1.6 sec
B = 4.2
6.3 = log(a/1.6) + 4.2
2.1 = log(a/1.6)
 now recall converting logs to exponents
10(2.1) = a/1.6
125.89 = a/1.6
201.4 microns = a
Therefore, the amplitude of the ground wave would be approx 201
microns
(C) Solving Simple Log Equations

Solve the following equations:
log 1 (81)  x
3
log
(
3
x

1
)


8
2
1
log 0.25 x (5)  
3
(D) Internet Links to Simple
Equations

Now work through the following
worksheet to reinforce your skills with
simple logarithms:

From edHelper.com - Logarithms
(E) Strategies for Solving Harder
Logarithmic Equations

the two key ideas in solving logarithmic equations are:

(1) get both sides of the equation to be a single
logarithmic expression
(2) both sides must have logarithms


the key tools you will use to work through logarithmic
equations are the three laws - product law, quotient law
and power law.
(E) Examples

Solve and verify and state the restrictions
for log3(x) - log3(4) = log3(12)

Solve and verify and state the restrictions
for log6(x + 3) + log6(x - 2) = 1

Solve and verify and state the restrictions
for log2(3x + 2) = 3 - log2(x - 1)
(F) Connections to Graphs

We will work out the
following examples on a
GDC:

Graph y1 = log(x) –
log(4)
Graph y2 = log(12)



Find their intersection
Which occurs when x =
48
(F) Connections to Graphs

But what happens when the base
of the logarithm is different???
The GDC works in base 10!!!!

We will use a different base (e) 
ln (loge) on a GDC:

Graph y1 = ln(x) – ln(4)
Graph y2 = ln(12)

Find their intersection ….
 Which occurs when x = 48
(although the y-value has
changed)  so base does NOT
matter for the solution of x

(F) Connections to Graphs




Recall that solving
equations also relates to
finding x-intercepts 
how does that apply
here?
If we rearrange the
equation, we get 
log(x) – log(4) – log(12)
=0
Now graph this
rearranged equation and
find the x-intercept(s)
(G) Internet Links

Solving Logarithmic Equations Lesson From PurpleMath

SOLVING LOGARITHMIC EQUATIONS from SOSMath

College Algebra Tutorial on Logarithmic
Equations from WTAMU
(D) Homework

Nelson text, page 146, Q3,6,7,8,12,16-19