A logarithm answers the question: to what power must a base be raised to produce a given positive number? The expression logₐ(x) = y means aʸ = x. For real logarithms, the base must be positive and not equal to 1, and the argument must be positive.

Main logarithm formulas

  • Product: logₐ(xy) = logₐ(x) + logₐ(y)
  • Quotient: logₐ(x/y) = logₐ(x) − logₐ(y)
  • Power: logₐ(xʳ) = r logₐ(x), when the real-valued expression is defined
  • Change of base: logₐ(x) = log_b(x) ÷ log_b(a)
  • Inverse relation: logₐ(aˣ) = x and a^(logₐ(x)) = x for valid inputs

Worked examples

Since 2³ = 8, log₂(8) = 3. Using the product rule, log₁₀(100 × 10) = log₁₀(100) + log₁₀(10) = 2 + 1 = 3. The change-of-base formula allows a calculator with common or natural logarithms to evaluate other bases.

How to apply a logarithm rule

  1. Check that every logarithm has a valid base and positive argument.
  2. Identify whether the expression involves a product, quotient, or power.
  3. Apply the matching identity without changing the base accidentally.
  4. Simplify the resulting terms and evaluate known logarithms.
  5. Substitute the result back into exponential form to check it.

Common errors and limitations

The product rule applies to multiplication inside a logarithm, not addition: logₐ(x + y) is not generally equal to logₐ(x) + logₐ(y). Likewise, the quotient rule concerns division. Logarithms of zero or negative real numbers are undefined in the real-number system, and a base of 1 is invalid.

Frequently asked questions

What is log base 10 of 100?

It is 2 because 10² = 100.

Can I use a different base?

Yes, provided the base is positive and not 1. Change of base can convert it to a base supported by your calculator.

Why must the argument be positive?

For real logarithms, a positive base raised to a real power is positive, so it cannot produce zero or a negative real number.