Logarithms: The "Inverse" of Exponents
Think of logarithms as a different way of looking at exponents. If an exponent tells you the result
of raising a base to a power, a logarithm tells you the power needed to reach a certain result.
1. The Core Definition
A logarithm asks the question: "To what power must we raise base b to get x?"
If you have an exponential equation:
𝑏𝑏 = 𝑏
The logarithmic equivalent is:
𝑏𝑏𝑏𝑏 (𝑏) = 𝑏
The "Loop" Trick: To convert a log back to an exponent, start at the base b, go across
the equals sign to y, and loop back to x.
Example:
𝑏𝑏𝑏2 8 = 3
because
3
2 =8
2. Common vs. Natural Logs
You will usually see two specific types of logs on your calculator:
Common Log: 𝑏𝑏𝑏10 (𝑏) is usually written as just 𝑏𝑏𝑏(𝑏). If there is no base written, assume
it is 10.
● Natural Log: 𝑏𝑏𝑏𝑏 (𝑏) is written as 𝑏𝑏(𝑏). It uses the mathematical constant e as a base
●
3. The Log Laws (The "Cheat Sheet")
Since logs are just exponents in disguise, they follow the same rules as Exponent Laws.
4. Why Do We Need Them?
Logarithms are the primary tool for solving exponential equations when you can’t get the same
base on both sides of the equation.
𝑏
Example: Solve 5 = 12
2. Use the Power Rule to bring 𝑏 down:
𝑏𝑏𝑏(5𝑏 ) = 𝑏𝑏𝑏(12)
𝑏 ⋅ 𝑏𝑏𝑏(5) = 𝑏𝑏𝑏(12)
3. Divide to isolate 𝑏:
𝑏=
4. Use your calculator to evaluate:
𝑏 ≈ 1.544
1. Take the log of both sides:
𝑏𝑏𝑏(12)
𝑏𝑏𝑏(5)
Pro Tip: Whenever you see a log, just rephrase it as a question: "What power do I put on the little
number to get the big number?" Your brain is already great at exponents—this just uses that same
skill in reverse!
Practice
1. Write each of the following in exponential notation
a) 𝑏𝑏𝑏3 81 = 4
d) 𝑏𝑏𝑏4
1
64
b) 𝑏𝑏𝑏7 7 = 1
c) 𝑏𝑏𝑏3 1 = 0
1
e) 𝑏𝑏𝑏 1 = 3
= −3
2
f) 𝑏𝑏𝑏𝑏 𝑏 = 𝑏
8
2. Write each of the following in logarithmic form
2
b) 4−2 =
a) 8 = 64
1 3
1
3
27
d) ( ) =
1 −5
1
c) ( )
16
2
1
= 32
f) 𝑏2𝑏 = 𝑏
e) 643 = 4
3. Find the value of y
1
a) 𝑏𝑏𝑏5 25 = 𝑏
b) 𝑏𝑏𝑏16 4 = 𝑏
c) 𝑏𝑏𝑏2 = 𝑏
d) 𝑏𝑏𝑏5 1 = 𝑏
e) 𝑏𝑏𝑏 1 9 = 𝑏
f) 𝑏𝑏𝑏8 2 = 𝑏
g) 𝑏𝑏𝑏𝑏 32 = 5
h) 𝑏𝑏𝑏𝑏
8
3
1
81
i) 𝑏𝑏𝑏9 𝑏 = −
= −4
1
2
4. Rewrite using the Power Rule
a) 𝑏𝑏𝑏2 𝑏3
b) 𝑏𝑏𝑏5 𝑏2
c) 𝑏𝑏𝑏3 𝑏5
d) 𝑏𝑏𝑏10 𝑏−4
5. Evaluate the value of each without a calculator
a) 𝑏𝑏𝑏2 8 + 𝑏𝑏𝑏3 27
b) 𝑏𝑏𝑏10 1000 + 𝑏𝑏𝑏5 25
c) 𝑏𝑏𝑏3
1
27
− 𝑏𝑏𝑏6 36
1
d) 𝑏𝑏𝑏 1 (2) + 𝑏𝑏𝑏4 ( )
2
64
e) 𝑏𝑏𝑏4 2 − 𝑏𝑏𝑏9 3
f) 𝑏𝑏𝑏7 1 + 𝑏𝑏𝑏8 8