6.4: Logarithmic Functions
Let’s compile a list of some inverse functions we know:
Function
𝑓(𝑥 ) = 𝑎𝑥
𝑓(𝑥 ) = √𝑥
𝑓(𝑥 ) = 𝑥 3
𝑎
𝑓 (𝑥 ) =
𝑥
Inverse
𝑥
𝑎
−1 ( )
2
𝑓 𝑥 =𝑥 ,𝑥 ≥0
3
𝑓 −1 (𝑥 ) = √𝑥
𝑎
𝑓 −1 (𝑥 ) = , 𝑥 ≠ 0
𝑓 −1 (𝑥 ) =
𝑥
𝑓(𝑥 ) = 𝑎 𝑥
????
We need a new operation!
1. Simplify each logarithmic expression:
a. log 2 8 =
b. log 2 32 =
c. log 3 27 =
1
d. log 5 =
5
e. log 4
1
16
=
So, logarithmic and exponential functions are inverses of each other.
That’ s pretty neat. Check this out:
https://www.desmos.com/calculator/ffsex0jnuz
Let’s look more closely at the graph of a logarithmic function:
In general, for f(x) = log 𝑎 𝑥
 contains the points: (1, 0),
1
(a, 1), and ( , -1)
𝑎
 has a vertical asymptote at
x=0
 is strictly increasing if a>1
 is strictly decreasing if 0<a<1
Domain of a logarithmic function / range of an exponential function:
Range of a logarithmic function / domain of the exponential function:
2.
Find the domain of each logarithmic function:
a. 𝑔(𝑥 ) = log 3 (2𝑥 + 3)
b. ℎ(𝑥 ) = log 9 (
1
𝑥−4
)
Let’s talk e:
So, log 𝑒 𝑥 = ln 𝑥
(“natural log of x”). Remember, 𝑒 ≈ 2.718 …
3.
Graph each function.
𝑟(𝑥 ) = 3 log 3 (𝑥 − 2)
𝑥
𝑝(𝑥 ) = ln ( ) + 1
2
Last definition, I promise
4.
Solve each logarithmic equation.
a. log 4 𝑥 = 3
b. log 5 625 = 𝑥 − 1
c. log 𝑥 32 = 5
d. log 1000 = 25𝑥
e. ln 𝑒 −2𝑥 = 8
f. 3𝑥+2 = 50
g. 2𝑒 𝑥−1 = 10
h. log 4 4𝑥 = −2
6.4 Homework:
p. 447: #9-113 EOO