Logarithm Formula

The Formula

b^y = x \implies \log_b(x) = y

When to use: The exponent that produces a number. \log_2(8) = 3 because 2^3 = 8.

Quick Example

\log_{10}(1000) = 3 (because 10^3 = 1000).
\log_2(16) = 4 (because 2^4 = 16).

Notation

\log_b(x) denotes the logarithm base b of x. \log usually means \log_{10}; \ln means \log_e.

What This Formula Means

The logarithm \log_b(x) answers: "to what power must b be raised to produce x?" It is the inverse function of f(x) = b^x.

The exponent that produces a number. \log_2(8) = 3 because 2^3 = 8.

Formal View

\log_b\colon (0,\infty) \to \mathbb{R} defined by \log_b(x) = y \iff b^y = x, where b > 0,\; b \neq 1

Worked Examples

Example 1

easy
Evaluate \log_2 32.

Solution

  1. 1
    A logarithm asks for the exponent, so we want the value of x such that 2^x = 32.
  2. 2
    Check powers of 2: 2^5 = 32.
  3. 3
    Therefore \log_2 32 = 5.

Answer

5
A logarithm answers the question: 'What power do I raise the base to in order to get this number?' The definition \log_b a = c means b^c = a.

Example 2

medium
Solve \log_3(2x + 1) = 4.

Example 3

hard
Solve \log_2(x) + \log_2(x - 6) = 4.

Common Mistakes

  • Thinking \log(a + b) = \log(a) + \log(b) — the log of a sum is NOT the sum of logs; only \log(ab) = \log(a) + \log(b)
  • Confusing \ln and \log — \ln is always base e; \log is usually base 10 (or context-dependent)
  • Forgetting that \log(0) and \log(\text{negative}) are undefined for real numbers

Why This Formula Matters

Undoes exponentials, measures orders of magnitude, appears in complexity analysis.

Frequently Asked Questions

What is the Logarithm formula?

The logarithm \log_b(x) answers: "to what power must b be raised to produce x?" It is the inverse function of f(x) = b^x.

How do you use the Logarithm formula?

The exponent that produces a number. \log_2(8) = 3 because 2^3 = 8.

What do the symbols mean in the Logarithm formula?

\log_b(x) denotes the logarithm base b of x. \log usually means \log_{10}; \ln means \log_e.

Why is the Logarithm formula important in Math?

Undoes exponentials, measures orders of magnitude, appears in complexity analysis.

What do students get wrong about Logarithm?

\log without a base usually means \log_{10} (common) or \log_e (natural).

What should I learn before the Logarithm formula?

Before studying the Logarithm formula, you should understand: exponential function, inverse function.

Want the Full Guide?

This formula is covered in depth in our complete guide:

Exponents and Logarithms: Rules, Proofs, and Applications →
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