2. Exponential Equations with Like Bases
ïșIn an Exponential Equation, the variable is in the
exponent. There may be one exponential term
or more than one, likeâŠ
ïșIf 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
ïșExample #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
ïșExample #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
weâll want to change to a
base so we can use our
calculator. That would be
either base 10 or base e.
LOG
âcommonâ
log base 10
LN
ânaturalâ 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
ïșThe Exponential Equations below contain
exponential expressions whose bases cannot
be rewritten as the same rational number.
ïșThe 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
ïșExample #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
ïșExample #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
ïșExample #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
ïșExample #2 - Two exponential expressions.
4. To isolate the variable,
we need to combine the âxâ
terms, then factor out the
âxâ 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
ïșIn 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 âŠ
log4 2x 1 3 5
lnx ln(2x 1) 1
logx 3 2
13. Logarithmic Equations
ïșExample 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 âxâ.
14. Logarithmic Equations
ïșExample 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 âeâ).
3. To solve for âxâ we must distribute
the âeâ and then collect the âxâ terms
together and factor out the âxâ and
divide.
x 0.613
15. Logarithmic Equations
ïșExample 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.