Section 3.4
Exponential &
Logarithmic Functions
Remember our properties . . .
Properties of Logarithms
For 𝑀 > 0 and 𝑁 > 0:
1. log 𝑏 𝑀𝑁 = log 𝑏 𝑀 + log 𝑏 𝑁
2.
𝑀
log 𝑏
𝑁
= log 𝑏 𝑀 − log 𝑏 𝑁
3. log 𝑏 𝑀𝑝 = 𝑝 ∙ log 𝑀
Product Rule
Quotient Rule
Power Rule
Change of Base Formula
• Can be used to compute logarithms that aren’t
base 10 or base 𝑒.
• Change of Base Formula:
log 𝑀 ln 𝑀
log 𝑀 𝑁 =
=
log 𝑁 ln 𝑁
Exponential Equations
• An exponential equation is an equation
containing a variable in an exponent.
• Ex’s: 23𝑥−8 = 16,
4𝑥 = 15, 40𝑒 0.6𝑥 = 240
• So, how do we solve exponential equations?
Example 1
Solve.
a. 53𝑥−6 = 125
b. 8𝑥+2 = 4𝑥−3
Summary of Method Used in Ex 1
• In Ex 1, we could rewrite both sides of the
equation using the same base.
What do we do here?
• Our previous method won’t work on these.
One Important Thing to Remember
• The logarithmic function is not defined for zero
or negative numbers.
• So, we need to check our solutions!
Example 2
Solve.
a. 4𝑥 = 15
b. 10𝑥 = 120000
Summary of Method Used in Ex 2
• This method is used when we can’t rewrite each
side using the same base.
Example 3
Solve.
40𝑒
0.6𝑥
− 3 = 237
Example 4
Solve.
5
𝑥−2
2𝑥+3
=4
Example 5
Solve.
𝑒
2𝑥
𝑥
− 4𝑒 + 3 = 0
Some Factoring Practice
a. 𝑒 2𝑥 − 4𝑒 𝑥 + 3 =
b.
𝑒 2𝑥 − 9 =
c.
4𝑒 2𝑥 − 12𝑒 𝑥 + 9 =
What about our logarithms?
• A logarithmic equation is an equation
containing a variable in a logarithmic expression.
• Ex’s:
log 4 (𝑥 + 3) = 2
and
ln(4𝑥 + 3) = ln
• Note: When solving logarithmic equations, we
always need to check our solution!
1
𝑥
Example 6
Solve.
log 2 (𝑥 − 4) = 3
Check.
Example 7
Solve.
4 ln 3𝑥 = 8
Check.
Example 8
log 2 𝑥 + log 2 (𝑥 − 7) = 3
Check:
Example 9
1
ln 𝑥 + 2 − ln(4𝑥 + 3) = ln
𝑥
Check:
Example 10
The formula 𝐴 = 36.1𝑒 0.0126𝑡 models the
population of California, 𝐴, in millions, 𝑡 years after
2005.
a. What was the population of California in 2005?
b. When will the population of California reach 40
million?
Example 11
How long will it take $25,000 to grow to
$500,000 at 9% annual interest compounded
monthly?
𝑟 𝑛𝑡
Don’t forget: 𝐴 = 𝑃 1 +
𝑛
Questions???
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