Introduction to Logarithms

Precalculus · Lesson 6.3 · 12 min read

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:

log_b(x) = y means exactly the same thing as b^y = 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
4^1/2 = 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 sidebar.

Properties of logarithms

Rule	Formula
Product rule	log_b(MN) = log_b(M) + log_b(N)
Quotient rule	log_b(M/N) = log_b(M) − log_b(N)
Power rule	log_b(M^p) = p · log_b(M)
Change of base	log_b(x) = log(x) / log(b)
Special values	log_b(1) = 0, log_b(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 2^x = 50.
log(2^x) = 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 5^x = 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

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