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.

The Formula

by=x  ⟹  log⁡b(x)=y

When to use: The exponent that produces a number. log⁡2(8)=3 because 23=8.

Quick Example

log⁡10(1000)=3 (because 103=1000).
log⁡2(16)=4 (because 24=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)=bx.

The exponent that produces a number. log⁡2(8)=3 because 23=8.

Formal View

log⁡b ⁣:(0,∞)→R defined by log⁡b(x)=y  ⟺  by=x, where b>0,  b≠1

Worked Examples

Example 1

easy
Evaluate log⁡232.

Answer

5

First step

1
A logarithm asks for the exponent, so we want the value of x such that 2x=32.

Full solution

  1. 2
    Check powers of 2: 25=32.
  2. 3
    Therefore log⁡232=5.
A logarithm answers the question: 'What power do I raise the base to in order to get this number?' The definition log⁡ba=c means bc=a.

Example 2

medium
Solve log⁡3(2x+1)=4.

Example 3

hard
Solve log⁡2(x)+log⁡2(x−6)=4.

Common Mistakes

  • Treating a log as division - log⁡b(x) is the exponent on b, not x÷b.
  • Taking the log of zero or a negative number - the argument of a log must be positive.
  • Confusing which way it inverts - a log frees the exponent (2x=8⇒x=log⁡28), a root frees the base.

Why This Formula Matters

Logarithms are the only clean way to solve equations where the unknown is an exponent, and they turn multiplication into addition, which is why they power slide rules, pH, and decibel scales. Treating log⁡b(x) as anything but 'the exponent' makes every log rule look arbitrary. Recognizing it by "Am I asking 'what exponent on the base gives this number?'" — rather than by familiar numbers — is what lets a student tell it apart from exponential function and root and natural logarithm in a mixed problem set.

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)=bx.

How do you use the Logarithm formula?

The exponent that produces a number. log⁡2(8)=3 because 23=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?

Logarithms are the only clean way to solve equations where the unknown is an exponent, and they turn multiplication into addition, which is why they power slide rules, pH, and decibel scales. Treating log⁡b(x) as anything but 'the exponent' makes every log rule look arbitrary. Recognizing it by "Am I asking 'what exponent on the base gives this number?'" — rather than by familiar numbers — is what lets a student tell it apart from exponential function and root and natural logarithm in a mixed problem set.

What do students get wrong about Logarithm?

The procedure for logarithm is the easy part; the trap is treating a log as division. Asking "Am I asking 'what exponent on the base gives this number?'" first is what keeps a correct-looking calculation from being attached to the wrong concept.

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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