A logarithm answers one question: what exponent do I need? log₂ 8 asks “2 to what power is 8?” The answer is 3, because 2³ = 8. Every log rule and every log equation comes back to that.
The definition
logb(x) = y means by = x.
The base b must be positive and not 1, and x must be positive. Being able to switch between the two forms is the most useful log skill there is.
| Log form | Exponential form | Value |
|---|---|---|
| log₂ 8 | 2³ = 8 | 3 |
| log₁₀ 1000 | 10³ = 1000 | 3 |
| log₅ (1/25) | 5−2 = 1/25 | −2 |
| log₄ 2 | 41/2 = 2 | ½ |
| logb 1 | b⁰ = 1 | 0 |
| logb b | b¹ = b | 1 |
Two bases get their own names. log x with no base written means base 10. ln x, the natural log, means base e, where e ≈ 2.718. Your calculator has a button for each.
The log rules
Because logs are exponents, they follow the exponent rules in a different outfit.
| Rule | Statement | Comes from |
|---|---|---|
| Product | logb(MN) = logb M + logb N | Multiply powers, add exponents |
| Quotient | logb(M/N) = logb M − logb N | Divide powers, subtract exponents |
| Power | logb(Mk) = k · logb M | Power of a power, multiply exponents |
| Change of base | logb x = ln x / ln b | Lets a calculator handle any base |
Example Expand log(x²y/z)
- Quotient rule: log(x²y) − log z.
- Product rule: log(x²) + log y − log z.
- Power rule: 2 log x + log y − log z.
Example Condense 3 log₂ x − log₂ 4
- Power rule: log₂(x³) − log₂ 4.
- Quotient rule: log₂(x³/4).
Example Find log₂ 10 with a calculator
Change of base: ln 10 / ln 2 ≈ 2.3026 / 0.6931 ≈ 3.32. Check: 23.32 ≈ 10. ✓
Solving exponential equations
When the variable is in the exponent, first try to write both sides with the same base. If you can’t, take a log of both sides.
Example Solve 5 · 2x = 40
- Divide by 5: 2x = 8.
- 8 = 2³, so x = 3.
Example Solve 3x = 20
- 20 isn’t a power of 3, so take ln of both sides: ln(3x) = ln 20.
- Power rule: x · ln 3 = ln 20.
- Divide: x = ln 20 / ln 3 ≈ 2.9957 / 1.0986 ≈ 2.73.
Solving log equations
Get a single log on one side, then rewrite in exponential form.
Example Solve log₃(x + 1) = 2
Rewrite: x + 1 = 3² = 9, so x = 8.
Example Solve log x + log(x − 3) = 1
- Product rule: log(x(x − 3)) = 1.
- Exponential form, base 10: x(x − 3) = 10.
- Rearrange: x² − 3x − 10 = 0, which factors to (x − 5)(x + 2) = 0.
- Candidates: x = 5 or x = −2.
- Check: log(−2) is undefined, so reject it. The answer is x = 5.
Always check the domain
You can only take the log of a positive number. Log equations often produce an extra solution that makes an argument negative or zero. Plug each answer back in and throw out any that fail.
Where logs show up
Logs are how you solve for time in growth and decay problems.
Example $1,000 earns 5% interest compounded yearly. How long until it doubles?
- Set up: 1000(1.05)t = 2000, so 1.05t = 2.
- Take ln: t · ln 1.05 = ln 2.
- t = ln 2 / ln 1.05 ≈ 0.6931 / 0.0488 ≈ 14.2 years.
Example Bacteria double every 3 hours. Starting from 500, when will there be 8,000?
8000/500 = 16 = 2⁴, so the population doubles 4 times. That takes 4 × 3 = 12 hours.
Logs also run the pH scale, the decibel scale for sound, and the Richter-style magnitude scales for earthquakes. Each step of 1 on those scales is a factor of 10.
Mistakes to avoid
- log(x + y) is not log x + log y. The product rule is about multiplying inside the log.
- log x / log y is not log(x/y). The quotient rule is a subtraction of logs.
- (log x)² is not 2 log x. The power rule needs the exponent inside the log.
Your turn
Evaluate log₃ 81.
Show answer
4, because 3⁴ = 81.
Evaluate log₂(1/32).
Show answer
−5, because 2−5 = 1/32.
Expand ln(x³/√y).
Show answer
3 ln x − ½ ln y. √y is y1/2.
Solve 2x+1 = 48.
Show answer
x = log₂ 24 ≈ 4.58. Divide by 2 to get 2x = 24, then x = ln 24 / ln 2.
Solve log₅(2x − 1) = 2.
Show answer
x = 13. 2x − 1 = 25.
Solve log₂ x + log₂(x + 2) = 3.
Show answer
x = 2. Combine to x(x + 2) = 8, so x² + 2x − 8 = 0 and (x + 4)(x − 2) = 0. Reject −4.
Practice
Questions students ask
Why is ln called natural?
The number e shows up whenever something grows continuously at a rate proportional to its size, from compound interest to population growth. Calculus with base e is also simpler than with any other base, which makes it the default in science.
Can a log be negative?
Yes. log₁₀ 0.01 = −2. What can’t be negative, or zero, is the number you take the log of.
How do logs connect to exponents?
They are inverse functions. logb(bx) = x and blogb x = x. Review the exponent rules if the log rules feel shaky.