Logarithm Rules: Product, Quotient, Power and Change of Base

A logarithm answers the question “what exponent do I need?”: log_b(x) = y means b^y = x. So log₂(8) = 3 because 2³ = 8. The core logarithm rules are the product rule (log xy = log x + log y), the quotient rule (log x/y = log x − log y), the power rule (log xⁿ = n log x) and the change-of-base rule (log_b x = ln x ÷ ln b). Each one is an exponent rule written in reverse.

The definition and the basic identities

Three examples make the definition concrete: log₁₀(1,000) = 3, log₂(32) = 5 and ln(e⁴) = 4. The common log (log, base 10) and the natural log (ln, base e ≈ 2.71828) are the two bases on every scientific calculator.

The three main rules

Product rule

Multiplying numbers adds their exponents (b^m × bⁿ = b^m+n), so the log of a product is the sum of the logs. Example: log 2 + log 50 = log 100 = 2.

Quotient rule

Dividing subtracts exponents, so the log of a quotient is a difference. Example: log₂(48) − log₂(3) = log₂(16) = 4.

Power rule

An exponent inside a log becomes a multiplier in front. This is the rule that lets you solve for an unknown exponent. Example: log(10⁶) = 6 log 10 = 6. Roots are fractional powers, so log √x = ½ log x.

Change of base

Calculators only have log and ln buttons. To evaluate a log in any other base, convert:

Either log or ln works, as long as you use the same one on top and bottom. The logarithm calculator evaluates any base directly.

Expanding and condensing logarithms

Expanding splits one log into several simpler ones:

Condensing runs the rules backward, power rule first:

Solving exponential equations

When the unknown is in an exponent, take the log of both sides and use the power rule to bring it down.

A growth example: how long does money take to double at 6% per year, compounded annually? Solve 1.06ᵗ = 2: t = ln 2 ÷ ln 1.06 ≈ 11.9 years. The familiar rule of 72 (72 ÷ 6 = 12) is a mental shortcut for this exact calculation. The solve for exponent calculator handles any equation of the form aˣ = b.

Solving logarithmic equations

Condense to a single log, rewrite in exponential form, solve, then check every answer, because the log of a zero or negative number is undefined.

Extraneous solutions like x = −2 appear often, so the checking step is not optional. The logarithm equation calculator flags them automatically.

Common mistakes

Logarithms in spreadsheets and code

Spreadsheets save you the change-of-base step. In Excel and Google Sheets, =LOG(50,3) returns log₃(50) ≈ 3.5609, =LOG10(1000) returns 3 and =LN(EXP(4)) returns 4. Leaving out the base, as in =LOG(100), defaults to base 10.

Programming languages differ, which trips up many people. In Python, JavaScript and C, the function named log is the natural logarithm, so Math.log(100) in JavaScript returns 4.605, not 2. Use log10 for base 10, log2 for base 2, or divide two logs for any other base.

A useful mental anchor: log 2 ≈ 0.301, log 3 ≈ 0.477 and log 5 ≈ 0.699. With the product and power rules, those three values give you the common log of many numbers without a calculator. For example, log 6 = log 2 + log 3 ≈ 0.778, and log 8 = 3 log 2 ≈ 0.903.

Where logarithms show up

Logarithmic scales compress huge ranges into manageable numbers, so each step of 1 means a tenfold change.

• pH is −log₁₀ of the hydrogen ion concentration. Lemon juice at pH 2 is about 100,000 times more acidic than pure water at pH 7. See how to calculate pH or the pH calculator.
• Decibels measure sound intensity as 10 log₁₀(I/I₀). Doubling the intensity adds 10 log 2 ≈ 3 dB. The decibel calculator does the conversions.
• Earthquake magnitude is logarithmic too: each whole step is ten times the measured amplitude and roughly 10^1.5 ≈ 32 times the energy released.

Because logarithms are the inverse of exponentials, every rule above pairs with one of the exponent rules. If one set feels unfamiliar, the other usually explains it.