Section 10.6
Exponential and Log Equations
To solve Exponential Equations you must first decide:
Can you get the same base on both sides as we did before
2x  1
4
x
2  12
2
x
2  2 −2
x  −2
or you cannot get the same base on both sides.
2x  7
If you cannot get the same base on both sides then we must use logarithms.
First isolate the exponential part on one side alone
2 x  7 is already isolated
5 x − 3  13 is not isolated and you should move the 3 to right hand side
5 x  16
Next identify the base in your problem
10 x  500 base is 10
2 x−3  12 base is 2
Now we need a fact from logarithms
if M  N
then log b M  log b N
10 x3  45
log 10 10 x3   log 10 45
Recall another fact
log b b x   x
log b M x   x ∗ log b M
Example 1
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2
A Scientific Report
10 x3  45
log 10 10 x3   log 10 45
x  3  log 10 45
x  −3  log 10 45
x  −1. 346 8
Example 2
2x  7
ln2 x   ln7
x ∗ ln2  ln7
ln2
x
ln7
x  0. 356 21
Example 3
3e 2x  1  16
3e 2x  15
e 2x  5
lne 2x   ln5
2x  ln5
ln5
x
2
x  0. 804
If the base is 10 use log10, if the base is e then use "ln" and if the base is not "e" or
"10" it does not matter what log you use.
Change of Base
You might have noticed that your calculator only has buttons for "log" and "ln" and
you wondered what do you do if you needed
log 5 125
Well have no fear we have a formula called Change of Base which allows you to use
logs that your calculator does have.
A Scientific Report
3
log b something
log b a
log 10 125
log 5 125 
3
log 10 5
ln125
3
log 5 125 
ln5
log a something) 
Logarithmic Equations
These equations can be condensed into 2 types:
Type 1
log b something  number
Type 2
log b something  log b something else
If your problem only has logs everywhere then it is type 2, but if it has a number
anywhere in it, it is type 1.
Solving Type 1
1.
2.
Use your properties of logs to collapse the problem
Use the basic fact that
log b something  number
b number  something
3.
4.
Check all of the answers since the domain for the log is x  0. Please note
that I am not saying throw away all negative answers.
Suppose your problem has a piece like
log 10 5 − x ...
and you get answers of
x  7, −2
then you would toss the 7 since it makes the INSIDE  0, but you would keep
the -2 since 5 − −2  positive.
Example Type 1
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log 2 5x − 3  5
2 5  5x − 3
32  5x − 3
35  5x
x7
Example Type 1
log 10 x  log 10 x  21  2
log 10 xx  21  2
10 2  xx  21
100  x 2  21x
x 2  21x − 100  0
x  25x − 4  0
x  −25, 4
x  4 and toss out the x  −25
Solving Type 2
1.
2.
Use your properties of logs to collapse the problem
Use the following fact
if log b M  log b N
then M  N
3.
Check all of the answers since the domain for the log is x  0.
Example Type 2
ln2t  5  ln3  lnt − 1
ln2t  5  ln3t − 1
2t  5  3t − 3
8t