# Notes

```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
1
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 &quot;ln&quot; and if the base is not &quot;e&quot; or
&quot;10&quot; it does not matter what log you use.
Change of Base
You might have noticed that your calculator only has buttons for &quot;log&quot; and &quot;ln&quot; 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 ...
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
4
A Scientific Report
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
```