Exponential & Logarithmic Functions

Properties of Logarithms

Using the product, quotient, and power properties to expand and condense logarithms.

Advanced20 min lesson3 min readUpdated August 12, 2026Author not yet attributed

Prerequisites

Introduction to Logarithms

Where Logarithm Rules Actually Come From

Before reading on, think back to the product rule for exponents: aᵐ · aⁿ = aᵐ⁺ⁿ — multiplying same-base powers adds their exponents. Since a logarithm literally is an exponent, predict what log_b(m · n) should equal in terms of log_b(m) and log_b(n) separately.

It should be a sum, and it is: log_b(m · n) = log_b(m) + log_b(n). This isn't a separate fact to memorize — it's the exact same product rule for exponents, just described in logarithm language instead of exponent language. Since logs are exponents, whatever exponent rules say about combining exponents, the matching logarithm properties say about combining logs.

Definition — Properties of Logarithms

Product property: log_b(mn) = log_b(m) + log_b(n). Quotient property: log_b(m/n) = log_b(m) − log_b(n). Power property: log_b(mᵖ) = p · log_b(m). Each follows directly from the matching exponent rule (product, quotient, and power rules).

Worked Example — Expanding a Logarithmic Expression

Expand log_2(8x³/y). Apply the quotient property first: log_2(8x³) − log_2(y). Apply the product property: log_2(8) + log_2(x³) − log_2(y). Apply the power property to the middle term: log_2(8) + 3log_2(x) − log_2(y). Since log_2(8) = 3: 3 + 3log_2(x) − log_2(y).

Worked Example — Condensing a Logarithmic Expression

Condense 2log_3(x) + log_3(y) − log_3(z) into a single logarithm. Apply the power property in reverse: log_3(x²) + log_3(y) − log_3(z). Apply the product and quotient properties in reverse: log_3(x²y/z).

\log_b(mn) = \log_b(m) + \log_b(n) \qquad \log_b(m^{p}) = p\log_b(m)

Tip

Whenever a logarithm property feels arbitrary, translate it back into its matching exponent rule — the product property mirrors aᵐ · aⁿ = aᵐ⁺ⁿ, and the power property mirrors (aᵐ)ⁿ = aᵐⁿ, since a repeated log addition is really a repeated exponent addition.

Common Mistakes

Treating log_b(m + n) as if it could be split into log_b(m) + log_b(n), confusing a sum inside a logarithm with a product inside one.
The product property applies to a product inside the logarithm, mn, not a sum, m + n — there's no rule for splitting apart the logarithm of a sum.
Forgetting to apply the power property when condensing an expression with a coefficient in front of a logarithm, such as leaving 2log_3(x) instead of converting it to log_3(x²) first.
A coefficient in front of a logarithm must become an exponent inside it before combining with other logarithms — 2log_3(x) becomes log_3(x²), then it can combine with the rest.

Key Takeaways

Every logarithm property is a direct translation of the matching exponent rule, since a logarithm is itself an exponent.
The product property turns a product inside a log into a sum of logs; the quotient property turns a quotient into a difference.
The power property moves an exponent inside a logarithm out front as a multiplying coefficient.

Summary

Logarithm properties aren't a separate set of rules — they're exponent rules wearing different notation. The final lesson in this unit uses these tools to actually solve equations where a variable is trapped in an exponent or inside a logarithm.

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