Logarithms answer a question
A logarithm answers: "To what exponent must I raise the base to get this number?"
log₂ 8 = 3, because 2³ = 8.
The definition
log_b x = y means bʸ = x
(b > 0, b ≠ 1, x > 0)
| Logarithmic form | Exponential form |
|---|---|
| log₂ 32 = 5 | 2⁵ = 32 |
| log₃ 81 = 4 | 3⁴ = 81 |
| log₁₀ 1,000 = 3 | 10³ = 1,000 |
| log₅ (1/25) = −2 | 5⁻² = 1/25 |
| log₄ 2 = ½ | 4^(1/2) = 2 |
Evaluating logarithms
Ask: base to what power gives the number?
- log₆ 36 = 2
- log₁₀ 0.01 = −2
- log₉ 3 = ½
- log₇ 1 = 0 (any base to the 0 is 1)
- log₈ 8 = 1
Special logarithms
| Type | Base | Written |
|---|---|---|
| Common logarithm | 10 | log x |
| Natural logarithm | e ≈ 2.718 | ln x |
Calculators have LOG and LN keys.
- log 100 = 2
- log 50 ≈ 1.699
- ln e = 1
Change of base
To find log₃ 20 on a calculator:
log_b x = log x ÷ log b
log₃ 20 = log 20 ÷ log 3 ≈ 2.727
Restrictions
- You can’t take the log of 0 or a negative number (no power of a positive base gives them).
- log_b x is negative when 0 < x < 1.
Inverse of exponentials
Logarithms undo exponentials: log_b(bˣ) = x and b^(log_b x) = x. The graphs of y = 2ˣ and y = log₂ x are reflections of each other in the line y = x.
Real-world logarithmic scales
| Scale | Measures | Each step of 1 means |
|---|---|---|
| Richter / moment magnitude | earthquakes | about 10× the wave amplitude (about 32× the energy) |
| Decibels | sound | +10 dB = 10× the intensity |
| pH | acidity | 10× change in hydrogen ion concentration |
Worked examples
Example 1: Write 2⁷ = 128 in logarithmic form: log₂ 128 = 7.
Example 2: Evaluate log₅ 125: 5³ = 125 → 3.
Example 3: Solve log₄ x = 3: x = 4³ = 64.
Common mistakes
- Mixing up the base and the answer: log₂ 8 = 3, not 8³.
- Trying to take log of a negative.
- Thinking log x means multiply. It’s a function.
Try it yourself
- Write 3⁵ = 243 in logarithmic form.
- Write log₇ 49 = 2 in exponential form.
- Evaluate log₂ (1/16).
- Evaluate log 0.001.
- Solve log_x 36 = 2.
Answers: 1) log₃ 243 = 5 2) 7² = 49 3) −4 4) −3 5) 6
Quick recap
- A logarithm is an exponent: log_b x = y ⇔ bʸ = x.
- Common logarithm (base 10) and natural logarithm (base e).
- Logarithms undo exponentials and power logarithmic scales.