AP PRECALCULUS — UNIT 2B · EXPONENTIAL & LOGARITHMIC FUNCTIONS
2.9 — Logarithmic Expressions
Notes — Inverting Exponentiation
💡 Learning Objectives (2.9.A)
By the end of this lesson you will be able to:
- Evaluate logarithmic expressions using the relationship between exponents and logarithms
- Convert between exponential form and logarithmic form
- Use the natural logarithm (base e) and the common logarithm (base 10) appropriately
- Recognize the inverse relationship between exponential and logarithmic operations
1. Where Logarithms Come From
Exponential equations such as 2^x = 32 are easy to solve when the answer is a whole number. But what about 2^x = 30? The value of x lies between 4 and 5, but no whole number works. We need a new operation that asks the question: ‘To what power must I raise a base b to obtain a given number y?’ That operation is the logarithm.
Definition: For a base b > 0 with b ≠ 1, the expression logb(y) is the exponent x such that b^x = y. In symbols:
log_b(y) = x if and only if b^x = y
2. Switching Between Forms
Every logarithmic statement is just an exponential statement written from a different angle. Being able to flip between the two forms is the most important skill in this lesson.
📘 Example — Form conversion
Logarithmic form ⇒ Exponential form
- log₂(8) = 3 ⇒ 2³ = 8
- log₅(1) = 0 ⇒ 5⁰ = 1
- log₁₀(0.01) = −2 ⇒ 10⁻² = 0.01
- log₃(1/9) = −2 ⇒ 3⁻² = 1/9
3. Two Special Bases
Although any positive base (other than 1) works, two bases come up so often that they have their own notation:
- Common logarithm: log(y) with no subscript means log10(y). Used heavily in scientific contexts (decibels, pH, Richter scale).
- Natural logarithm: ln(y) means loge(y), where e ≈ 2.71828… Used whenever continuous growth or decay is involved.
⚠️ Common mistake
‘log’ on most calculators is base 10, while ‘ln’ is base e. There is no ‘log_b’ button — you will learn the change-of-base formula in 2.12 to evaluate other bases.
4. Evaluating Logarithms by Inspection
When the input is a recognizable power of the base, you can evaluate the logarithm without a calculator. Ask yourself: ‘What exponent on the base produces this number?’
📘 Example — Evaluation by inspection
- log₂(64) = 6 because 2⁶ = 64
- log₄(2) = 1/2 because 4^(1/2) = 2
- log₇(7) = 1 because 7¹ = 7
- ln(e³) = 3 because e³ = e³
- log(1000) = 3 because 10³ = 1000
5. Domain of a Logarithm
Because b^x is always positive when b > 0, the equation b^x = y has no solution for y ≤ 0. Therefore log_b(y) is only defined when y > 0.
⚠️ Common mistake
Expressions like log(0) and log(−5) are undefined. When you see log(x − 3), the argument x − 3 must be strictly positive, so the domain is x > 3.
6. A Numerical Look at log₂(y)
The table below shows pairs (y, log₂(y)). Notice how outputs grow slowly as y grows: doubling y always adds exactly 1 to the logarithm.
y | 1 | 2 | 4 | 8 | 16 | 32 |
|---|---|---|---|---|---|---|
log₂(y) | 0 | 1 | 2 | 3 | 4 | 5 |
Each time y doubles, log₂(y) goes up by 1. This compressing behavior is what makes logarithms ideal for handling quantities that span many orders of magnitude.
7. Summary
- log_b(y) = x is the exponent x to which the base b must be raised to give y
- Logarithmic and exponential forms are interchangeable: log_b(y) = x ⇔ b^x = y
- ‘log’ without a subscript means base 10; ‘ln’ means base e
- The argument of any logarithm must be positive
- Evaluate by asking: ‘What exponent on the base gives this number?’