This document discusses different types of exponential and logarithmic equations and methods for solving them. It covers: 1) Exponential equations with like bases, which can be solved by setting exponents equal to each other. 2) Exponential equations with different bases, which require using logarithms and properties of logarithms to isolate the variable. 3) Logarithmic equations, where the variable can be inside or outside the logarithm. These also use logarithm properties and are rewritten as exponential equations to solve for the variable.

1. Exponential & Logarithmic
Equations
• Exponential Equations with Like
Bases
• Exponential Equations with Different
Bases
• Logarithmic Equations

2. Exponential Equations with Like Bases
In an Exponential Equation, the variable is in the
exponent. There may be one exponential term
or more than one, like…
If you can isolate terms so that the equation can
be written as two expressions with the same
base, as in the equations above, then the
solution is simple.
32x 1
5 4 or 3x 1
9x 2

3. Exponential Equations with Like Bases
Example #1 - One exponential expression.
32x 1
5 4
32x 1
9
32x 1
32
1. Isolate the exponential
expression and rewrite the
constant in terms of the same
base.
2x 1 2
2x 1
x
1
2
2. Set the exponents equal to
each other (drop the bases) and
solve the resulting equation.

4. Exponential Equations with Like Bases
Example #2 - Two exponential expressions.
3x 1
9x 2
3x 1
32 x 2
3x 1
32x 4
1. Isolate the exponential
expressions on either side of the
=. We then rewrite the 2nd
expression in terms of the same
base as the first.
2. Set the exponents equal to
each other (drop the bases) and
solve the resulting equation.
x 1 2x 4
x 5

5. Change-of-Base Formula
The base you change to can
be any base so generally
we’ll want to change to a
base so we can use our
calculator. That would be
either base 10 or base e.
LOG
“common”
log base 10
LN
“natural” log
base e
a
M
log
log
a
M
ln
ln
Example
for TI-83
If we generalize the process we just did
we come up with the:
a
M
M
b
b
a
log
log
log

6. Use the Change-of-Base Formula and a calculator to
approximate the logarithm. Round your answer to three
decimal places.
16log3
Since 32 = 9 and 33 = 27, our answer of what exponent
to put on 3 to get it to equal 16 will be something
between 2 and 3.
3ln
16ln
16log3
put in calculator
524.2

7. Exponential Equations with Different
Bases
The Exponential Equations below contain
exponential expressions whose bases cannot
be rewritten as the same rational number.
The solutions are irrational numbers, we will
need to use a log function to evaluate them.
32x 1
5 11 or 3x 1
4x 2

8. Exponential Equations with Different
Bases
Example #1 - One exponential expression.
1. Isolate the exponential
expression.
3. Use the log rule that lets you
rewrite the exponent as a
multiplier.
32x 1
5 11
32x 1
16
2. Take the log (log or ln) of both
sides of the equation.ln 32x 1
ln 16
(2x 1)ln3 ln16

9. Exponential Equations with Different
Bases
Example #1 - One exponential expression.
4. Isolate the variable.
2x 1
ln16
ln 3
2x
ln16
ln 3
1
x
ln16
2ln 3
1
2
x 0.762
(2x 1)ln3 ln16

10. Exponential Equations with Different
Bases
Example #2 - Two exponential expressions.
1. The exponential expressions
are already isolated.
3. Use the log rule that lets you
rewrite the exponent as a
multiplier on each side..
2. Take the log (log or ln) of both
sides of the equation.
3x 1
4x 2
ln 3x 1
ln 4x 2
(x 1)ln3 (x 2)ln4

11. Exponential Equations with Different
Bases
Example #2 - Two exponential expressions.
4. To isolate the variable,
we need to combine the ‘x’
terms, then factor out the
‘x’ and divide.
xln3 ln3 xln4 2ln4
xln3 xln4 ln3 2ln4
x(ln3 ln4) (ln3 2ln4)
x
(ln3 2ln4)
ln3 ln4
x 13.457

12. Logarithmic Equations
In a Logarithmic Equation, the variable can be
inside the log function or inside the base of the
log. There may be one log term or more than
one. For example …
log4 2x 1 3 5
lnx ln(2x 1) 1
logx 3 2

13. Logarithmic Equations
Example 1 - Variable inside the log function.
log4 2x 1 3 5
log4 2x 1 2
42
2x 1
16 2x 1
2x 17
x 8.5
1. Isolate the log expression.
2. Rewrite the log equation as
an exponential equation and
solve for ‘x’.

14. Logarithmic Equations
Example 2 - Variable inside the log function, two
log expressions.
ln x ln(2x 1) 1
ln
x
2x 1
1
e1 x
2x 1
e(2x 1) x
2ex e x
2ex x e
x(2e 1) e
x
e
2e 1
1. To isolate the log expression, we
1st must use the log property to
combine a difference of logs.
2. Rewrite the log equation as an
exponential equation (here, the
base is ‘e’).
3. To solve for ‘x’ we must distribute
the ‘e’ and then collect the ‘x’ terms
together and factor out the ‘x’ and
divide.
x 0.613

15. Logarithmic Equations
Example 3 - Variable inside the base of the log.
logx 3 2
x 2
3
x 2
1
2
3
1
2
x
1
3
x
3
3
1. Rewrite the log equation as
an exponential equation.
2. Solve the exponential
equation.