Logarithm

Q: What is the difference between a common logarithm and a natural logarithm?

A common logarithm uses base 10 and is written $\log x$ (or $\log_{10} x$). A natural logarithm uses the special base $e \approx 2.718$ and is written $\ln x$. Both follow the same rules; they differ only in their base. Calculators typically have separate keys for each.

Q: How do you convert between logarithmic and exponential form?

The statement $\log_b x = y$ is exactly equivalent to $b^y = x$. To convert, identify the base $b$, the exponent $y$ (the logarithm's value), and the result $x$ (the argument). Moving between these two forms is the single most important skill when working with logarithms.

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 log_b x. For example, log₂8 equals 3 since 2³ = 8.

Key Formula

\log_b x = y \quad \Longleftrightarrow \quad b^{\,y} = x

Where:

$b$ = The base of the logarithm (must be positive and not equal to 1)
$x$ = The argument (the number you are taking the logarithm of; must be positive)
$y$ = The logarithm's value — the exponent to which b must be raised to get x

Worked Example

Problem: Evaluate

\log_5 125

Step 1: Set up the equivalent exponential equation. Let the unknown logarithm equal y.

\log_5 125 = y \quad \Longleftrightarrow \quad 5^{\,y} = 125

Step 2: Express 125 as a power of 5.

125 = 5^3

Step 3: Since the bases match, the exponents must be equal.

5^{\,y} = 5^3 \;\Longrightarrow\; y = 3

Answer:

\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

x

\log_3 x = 4

Step 1: Rewrite the logarithmic equation in exponential form.

\log_3 x = 4 \quad \Longleftrightarrow \quad 3^4 = x

Step 2: Compute the power.

3^4 = 81

Step 3: State the solution.

x = 81

Answer:

x = 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

(or

\log_{10} x

). A natural logarithm uses the special base

e \approx 2.718

and is written

\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

y

for which

10^y = -5

10^y = 0

. This is why the domain of

\log_b x

x > 0

How do you convert between logarithmic and exponential form?

The statement

\log_b x = y

is exactly equivalent to

b^y = x

. To convert, identify the base

b

, the exponent

y

(the logarithm's value), and the result

x

(the argument). Moving between these two forms is the single most important skill when working with logarithms.

Logarithm vs. Exponent

	Logarithm	Exponent
Definition	The exponent needed to raise a base to get a number	The power to which a base is raised
Notation	$\log_b x = y$	$b^y = x$
What it finds	The unknown exponent, given the base and result	The result, given the base and exponent
Relationship	Inverse of exponentiation	Inverse 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

is one of the core functions you differentiate and integrate.

Common Mistakes

Mistake: Thinking

\log_b(x + y) = \log_b x + \log_b y

Correction: The product rule states

\log_b(x \cdot y) = \log_b x + \log_b y

. There is no simple rule for

\log_b(x + y)

. Confusing addition inside the argument with multiplication is one of the most frequent logarithm errors.

Mistake: Forgetting that

\log_b 1 = 0

and

\log_b b = 1

Correction: Since

b^0 = 1

, the logarithm of 1 in any base is always 0. Since

b^1 = b

, the logarithm of the base itself is always 1. Keeping these two benchmarks in mind helps you check your work quickly.

Related Terms

Base of a Logarithm — The number repeatedly multiplied in a logarithm
Power — A logarithm finds the unknown power
Logarithm Rules — Properties for simplifying logarithmic expressions
Common Logarithm — Logarithm with base 10
Natural Logarithm — Logarithm with base e ≈ 2.718
Change of Base Formula — Converts between different logarithm bases
Exponential Function — The inverse operation of a logarithm