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Logarithm — Definition, Formula & Examples

Logarithm

The logarithm base b of a number x is the power to which b must be raised in order to equal x. This is written logb x. For example, log2 8 equals 3 since 23 = 8.

See also

Logarithm rules, common logarithm, natural logarithm, change of base formula

Key Formula

log⁡bx=y⟺b y=x\log_b x = y \quad \Longleftrightarrow \quad b^{\,y} = x
Where:
  • bb = The base of the logarithm (must be positive and not equal to 1)
  • xx = The argument (the number you are taking the logarithm of; must be positive)
  • yy = The logarithm's value — the exponent to which b must be raised to get x

Worked Example

Problem: Evaluate log⁡5125\log_5 125.
Step 1: Set up the equivalent exponential equation. Let the unknown logarithm equal y.
log⁡5125=y⟺5 y=125\log_5 125 = y \quad \Longleftrightarrow \quad 5^{\,y} = 125
Step 2: Express 125 as a power of 5.
125=53125 = 5^3
Step 3: Since the bases match, the exponents must be equal.
5 y=53  ⟹  y=35^{\,y} = 5^3 \;\Longrightarrow\; y = 3
Answer: log⁡5125=3\log_5 125 = 3

Another Example

The first example evaluates a logarithm directly; this example reverses the task by solving for the argument inside the logarithm, reinforcing the conversion between logarithmic and exponential form.

Problem: Solve for xx: log⁡3x=4\log_3 x = 4.
Step 1: Rewrite the logarithmic equation in exponential form.
log⁡3x=4⟺34=x\log_3 x = 4 \quad \Longleftrightarrow \quad 3^4 = x
Step 2: Compute the power.
34=813^4 = 81
Step 3: State the solution.
x=81x = 81
Answer: x=81x = 81

Frequently Asked Questions

What is the difference between a common logarithm and a natural logarithm?
A common logarithm uses base 10 and is written log⁡x\log x (or log⁡10x\log_{10} x). A natural logarithm uses the special base e≈2.718e \approx 2.718 and is written ln⁡x\ln x. Both follow the same rules; they differ only in their base. Calculators typically have separate keys for each.
Why can you only take the logarithm of a positive number?
Because no real exponent can make a positive base produce zero or a negative result. For instance, there is no real yy for which 10y=−510^y = -5 or 10y=010^y = 0. This is why the domain of log⁡bx\log_b x is x>0x > 0.
How do you convert between logarithmic and exponential form?
The statement log⁡bx=y\log_b x = y is exactly equivalent to by=xb^y = x. To convert, identify the base bb, the exponent yy (the logarithm's value), and the result xx (the argument). Moving between these two forms is the single most important skill when working with logarithms.

Logarithm vs. Exponent

LogarithmExponent
DefinitionThe exponent needed to raise a base to get a numberThe power to which a base is raised
Notationlog⁡bx=y\log_b x = yby=xb^y = x
What it findsThe unknown exponent, given the base and resultThe result, given the base and exponent
RelationshipInverse of exponentiationInverse of taking a logarithm

Why It Matters

Logarithms appear throughout algebra, precalculus, and science whenever you need to "undo" an exponential. They are essential for solving exponential growth and decay problems — from compound interest calculations to radioactive half-life equations. You will also use them heavily in calculus, where ln⁡x\ln x is one of the core functions you differentiate and integrate.

Common Mistakes

Mistake: Thinking log⁡b(x+y)=log⁡bx+log⁡by\log_b(x + y) = \log_b x + \log_b y.
Correction: The product rule states log⁡b(x⋅y)=log⁡bx+log⁡by\log_b(x \cdot y) = \log_b x + \log_b y. There is no simple rule for log⁡b(x+y)\log_b(x + y). Confusing addition inside the argument with multiplication is one of the most frequent logarithm errors.
Mistake: Forgetting that log⁡b1=0\log_b 1 = 0 and log⁡bb=1\log_b b = 1.
Correction: Since b0=1b^0 = 1, the logarithm of 1 in any base is always 0. Since b1=bb^1 = b, the logarithm of the base itself is always 1. Keeping these two benchmarks in mind helps you check your work quickly.

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