Properties of Logarithms: Sum, Difference, Power

The three main properties of logarithms are the product rule log_b(xy) = log_b x + log_b y, the quotient rule log_b(x/y) = log_b x − log_b y and the power rule log_b(xⁿ) = n log_b x. In brief: log₂ 12 = log₂ 3 + log₂ 4 = log₂ 3 + 2, log₁₀ (1000/8) = 3 − log₁₀ 8 and ln(x⁵) = 5 ln x. These rules correspond to the exponent laws.

The properties turn multiplication into addition, division into subtraction and powers into multiplication. They are used to expand and combine logarithmic expressions, to solve equations and to simplify calculations. This guide proves the three rules from the exponent laws, shows how to expand and condense expressions, explains the special cases and gives a table of the rules with examples.

What are the logarithm rules?

For b > 0, b ≠ 1, x > 0 and y > 0:

• Product: log_b(xy) = log_b x + log_b y.
• Quotient: log_b(x/y) = log_b x − log_b y.
• Power: log_b(xⁿ) = n log_b x for any real n.
• Reciprocal: log_b(1/x) = −log_b x.
• Root: log_b(ⁿ√x) = (1/n) log_b x.

Numerical check: log₁₀ 2 + log₁₀ 5 = 0.3010 + 0.6990 = 1 = log₁₀ 10 ✓. And log₁₀ 1000 − log₁₀ 10 = 3 − 1 = 2 = log₁₀ 100 ✓. And log₂ 8² = log₂ 64 = 6 = 2 · 3 ✓.

Why do the rules hold?

Let m = log_b x and n = log_b y, so x = b^m and y = bⁿ. Then xy = b^m · bⁿ = b^m+n. Therefore log_b(xy) = m + n, which is the product rule. For the quotient, x/y = b^m−n. For the power, x^k = (b^m)^k = b^mk, so log_b(x^k) = k m.

The rules hold for arguments that are positive. Note that log_b(xy) = log_b x + log_b y requires both x and y to be positive, but xy is positive also when both are negative. Then use absolute values: log(xy) = log|x| + log|y|.

How do you expand and condense logarithms?

Expanding

Write a single logarithm as a sum and difference of simpler ones.

Example 1

log₃(9x²/√y) = log₃ 9 + 2 log₃ x − (1/2) log₃ y = 2 + 2 log₃ x − (1/2) log₃ y.

Example 2

ln(x(x + 1)³/(x − 2)) = ln x + 3 ln(x + 1) − ln(x − 2). Note that ln(x + 1) cannot be expanded further, since the argument is a sum.

Condensing

Write a sum and difference as a single logarithm: first move coefficients to exponents, then combine sums into products and differences into quotients.

Example 3

2 log 5 + log 4 − log 10 = log 25 + log 4 − log 10 = log(25 · 4/10) = log 10 = 1.

Example 4

3 ln x − (1/2) ln y + ln 2 = ln(2x³/√y).

How do you use the rules to solve problems?

Example 5: computing with known logarithms

Given log₁₀ 2 ≈ 0.3010 and log₁₀ 3 ≈ 0.4771. Then log 6 = 0.3010 + 0.4771 = 0.7781; log 12 = 2 · 0.3010 + 0.4771 = 1.0791; log 1.5 = 0.4771 − 0.3010 = 0.1761; log 5 = 1 − 0.3010 = 0.6990 (since 5 = 10/2); and log 0.25 = −2 · 0.3010 = −0.6020.

Example 6: solving an equation

log₂ x + log₂(x − 2) = 3. Combine: log₂(x(x − 2)) = 3, so x(x − 2) = 8, which gives x² − 2x − 8 = 0, so (x − 4)(x + 2) = 0. The domain needs x > 2, so x = −2 is rejected. The solution is x = 4. Check: log₂ 4 + log₂ 2 = 2 + 1 = 3 ✓.

Example 7: solving for an exponent

3^x = 20. Take log: x log 3 = log 20, so x = log 20/log 3 = 1.3010/0.4771 = 2.727.

Example 8: decibels

If the intensity of a sound doubles, the level changes by 10 log₁₀ 2 ≈ 3.01 dB. If the intensity is multiplied by 1000, the change is 30 dB. The addition of decibels corresponds to the multiplication of intensities by the product rule.

Rules summary table

Where do the rules come from?

The product rule is the reason why logarithms were invented: Napier’s tables allowed astronomers to replace long multiplications by additions. Before logarithms, a method called prosthaphaeresis, based on trigonometric identities, was used for the same purpose. The rules were used for generations in the form of the slide rule, which adds lengths on logarithmic scales.

Where are they used today?

In data analysis, taking logarithms turns multiplicative relationships into additive ones, so that growth rates become slopes. Information theory measures the information in bits by log₂, and the information of independent events adds, which is the product rule at work. Logarithmic scales on graphs make exponential growth appear as a straight line.

Which mistakes are common?

Using the product rule for a sum

log(x + y) is not log x + log y. For x = y = 10: log 20 = 1.301, but log 10 + log 10 = 2.

Writing log(x/y) as log x / log y

The quotient rule gives a difference: log x − log y.

Applying the power rule to the wrong part

log(x²) = 2 log x, but (log x)² is a different number: log(100) = 2, while (log 100)² = 4.

Ignoring the domain

After combining logarithms, check that every original argument is positive for your solution.