Logarithm Formula

The logarithm _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=xโ€…โ€ŠโŸนโ€…โ€Šlogโกb(x)=yb^y = x \implies \log_b(x) = y

When to use: The exponent that produces a number. logโก2(8)=3\log_2(8) = 3 because 23=82^3 = 8.

Quick Example

logโก10(1000)=3\log_{10}(1000) = 3 (because 103=100010^3 = 1000).
logโก2(16)=4\log_2(16) = 4 (because 24=162^4 = 16).

Notation

logโกb(x)\log_b(x) denotes the logarithm base bb of xx. logโก\log usually means logโก10\log_{10}; lnโก\ln means logโกe\log_e.

What This Formula Means

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

The exponent that produces a number. logโก2(8)=3\log_2(8) = 3 because 23=82^3 = 8.

Formal View

logโกbโ€‰โฃ:(0,โˆž)โ†’R\log_b\colon (0,\infty) \to \mathbb{R} defined by logโกb(x)=yโ€…โ€ŠโŸบโ€…โ€Šby=x\log_b(x) = y \iff b^y = x, where b>0,โ€…โ€Šbโ‰ 1b > 0,\; b \neq 1

Worked Examples

Example 1

easy
Evaluate logโก232\log_2 32.

Answer

55

First step

1
A logarithm asks for the exponent, so we want the value of xx such that 2x=322^x = 32.

Full solution

  1. 2
    Check powers of 2: 25=322^5 = 32.
  2. 3
    Therefore logโก232=5\log_2 32 = 5.
A logarithm answers the question: 'What power do I raise the base to in order to get this number?' The definition logโกba=c\log_b a = c means bc=ab^c = a.

Example 2

medium
Solve logโก3(2x+1)=4\log_3(2x + 1) = 4.

Example 3

hard
Solve logโก2(x)+logโก2(xโˆ’6)=4\log_2(x) + \log_2(x - 6) = 4.

Common Mistakes

  • Treating a log as division - logโกb(x)\log_b(x) is the exponent on bb, not xรทbx\div 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=8โ‡’x=logโก282^x=8\Rightarrow x=\log_2 8), 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 logโกb(x)\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?'" โ€” rather than by familiar numbers โ€” 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 logโกb(x)\log_b(x) answers: "to what power must bb be raised to produce xx?" It is the inverse function of f(x)=bxf(x) = b^x.

How do you use the Logarithm formula?

The exponent that produces a number. logโก2(8)=3\log_2(8) = 3 because 23=82^3 = 8.

What do the symbols mean in the Logarithm formula?

logโกb(x)\log_b(x) denotes the logarithm base bb of xx. logโก\log usually means logโก10\log_{10}; lnโก\ln means logโกe\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 logโกb(x)\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?'" โ€” rather than by familiar numbers โ€” 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 โ†’
โ† Back to Logarithm See All Examples Practice Problems