A logarithm answers a simple question: what power do I need to raise a base to, to get a certain number? Logarithms are the inverse of exponential functions, and they show up everywhere from measuring earthquakes and sound to calculating compound interest.
What is a logarithm?
For a base b > 0 with b ≠ 1, and any x > 0:
logb(x) = y means exactly the same thing as by = x
In words: the logarithm of x with base b is the exponent you put on b to get x.
Example 1. Evaluate log2(8).
Ask: 2 to what power is 8? Since 23 = 8, we get log2(8) = 3.
Example 2. Evaluate log5(1/25).
Since 5−2 = 1/25, we get log5(1/25) = −2.
Converting between forms
Being able to switch between exponential form and logarithmic form is the most important skill in this unit.
| Exponential form | Logarithmic form |
|---|
| 34 = 81 | log3(81) = 4 |
| 10−1 = 0.1 | log10(0.1) = −1 |
| e0 = 1 | ln(1) = 0 |
| 41/2 = 2 | log4(2) = 1/2 |
Common and natural logarithms
Two bases are used so often that they get their own notation:
- The common logarithm uses base 10 and is written log(x). For example, log(1000) = 3.
- The natural logarithm uses base e ≈ 2.71828 and is written ln(x). For example, ln(e5) = 5.
Most calculators have a log key and an ln key for these. Try them on the calculator in the Tools box.
Properties of logarithms
| Rule | Formula |
|---|
| Product rule | logb(MN) = logb(M) + logb(N) |
| Quotient rule | logb(M/N) = logb(M) − logb(N) |
| Power rule | logb(Mp) = p · logb(M) |
| Change of base | logb(x) = log(x) / log(b) |
| Special values | logb(1) = 0, logb(b) = 1 |
Example 3. Expand log2(8x3).
log2(8) + log2(x3) = 3 + 3·log2(x)
Example 4. Use change of base to find log3(20).
log3(20) = log(20) / log(3) ≈ 1.3010 / 0.4771 ≈ 2.727
Solving exponential equations
When the variable is in the exponent, take the logarithm of both sides and use the power rule to bring it down.
Example 5. Solve 2x = 50.
log(2x) = log(50) ⇒ x·log(2) = log(50) ⇒ x = log(50)/log(2) ≈ 5.644
Practice problems
- Evaluate log3(27).
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3
- Evaluate log(0.01).
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−2
- Rewrite 62 = 36 in logarithmic form.
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log6(36) = 2
- Rewrite log7(49) = 2 in exponential form.
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72 = 49
- Expand log(100x/y).
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2 + log(x) − log(y)
- Condense 2·ln(a) + ln(b).
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ln(a2b)
- Solve 5x = 200. Round to three decimal places.
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x = log(200)/log(5) ≈ 3.292
- Solve log2(x + 3) = 5.
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x + 3 = 32, so x = 29