Base of a Logarithm — Definition, Formula & Examples

Q: What is the base of a logarithm when no base is written?

When you see $\log x$ with no subscript, the base is assumed to be 10. This is called the common logarithm. On many calculators the 'log' button uses base 10 by default. However, the notation $\ln x$ denotes the natural logarithm, which has base $e \approx 2.718$.

Q: Why can't the base of a logarithm be 1 or negative?

If the base were 1, then $1^y = 1$ for every exponent $y$, so the logarithm could never produce any value other than 1 — it would be undefined for all other inputs. If the base were negative, raising it to non-integer exponents would produce complex (non-real) numbers, which breaks the real-valued definition of logarithms.

Q: How do you change the base of a logarithm?

Use the change of base formula: $\log_b x = \frac{\log_c x}{\log_c b}$, where $c$ is any convenient base. Most often you choose $c = 10$ or $c = e$ so you can evaluate the expression on a calculator. For example, $\log_2 32 = \frac{\log_{10} 32}{\log_{10} 2} = \frac{1.505}{0.301} = 5$.

Base of a Logarithm

For log_bx, the base is b. Similar to the base of an exponential expression.

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 (b > 0, b ≠ 1).
$x$ = The argument (input) of the logarithm. Must be positive (x > 0).
$y$ = The exponent — the power to which b must be raised to equal x.

Worked Example

Problem: Evaluate

\log_2 32

Step 1: Identify the base. Here the base is 2.

\log_{\mathbf{2}}\, 32

Step 2: Rewrite the logarithmic equation in exponential form. You need the exponent y such that the base raised to y equals 32.

2^y = 32

Step 3: Express 32 as a power of 2.

32 = 2^5

Step 4: Since the bases match, set the exponents equal.

y = 5

Answer:

\log_2 32 = 5

, because the base 2 must be raised to the 5th power to produce 32.

Another Example

This example works in reverse: instead of evaluating the logarithm, you solve for the unknown base itself.

Problem: Solve for the base

b

\log_b 81 = 4

Step 1: Convert the logarithmic equation to exponential form using the definition.

b^4 = 81

Step 2: Express 81 as a perfect fourth power. Since

3^4 = 81

, the base must be 3.

81 = 3^4

Step 3: Match the expressions to find the base.

b^4 = 3^4 \implies b = 3

Answer: The base is

b = 3

, because

3^4 = 81

Frequently Asked Questions

What is the base of a logarithm when no base is written?

When you see

\log x

with no subscript, the base is assumed to be 10. This is called the common logarithm. On many calculators the 'log' button uses base 10 by default. However, the notation

\ln x

denotes the natural logarithm, which has base

e \approx 2.718

Why can't the base of a logarithm be 1 or negative?

If the base were 1, then

1^y = 1

for every exponent

y

, so the logarithm could never produce any value other than 1 — it would be undefined for all other inputs. If the base were negative, raising it to non-integer exponents would produce complex (non-real) numbers, which breaks the real-valued definition of logarithms.

How do you change the base of a logarithm?

Use the change of base formula:

\log_b x = \frac{\log_c x}{\log_c b}

, where

c

is any convenient base. Most often you choose

c = 10

c = e

so you can evaluate the expression on a calculator. For example,

\log_2 32 = \frac{\log_{10} 32}{\log_{10} 2} = \frac{1.505}{0.301} = 5

Base of a Logarithm vs. Base of an Exponential Expression

	Base of a Logarithm	Base of an Exponential Expression
Definition	The number b in log_b(x) that serves as the repeated factor	The number b in b^y that is raised to a power
Notation	Written as a subscript: log_b	Written before the exponent: b^y
Restrictions	b > 0 and b ≠ 1	b > 0 and b ≠ 1 (for the corresponding logarithm to exist)
Relationship	log_b(x) = y asks 'what exponent gives x?'	b^y = x states the result of raising b to y
Common defaults	Base 10 (common log) or base e (natural log)	Base 10 (powers of 10) or base e (exponential growth)

Why It Matters

The base of a logarithm appears throughout algebra, precalculus, and science whenever you need to 'undo' an exponential. Choosing the right base determines the scale of measurement — base 10 is used in the Richter scale and pH, while base

e

underlies continuous growth models in biology and finance. Understanding the base is also essential for applying logarithm rules correctly and for converting between bases with the change of base formula.

Common Mistakes

Mistake: Confusing the base with the argument. For example, reading

\log_3 81

and thinking 81 is the base.

Correction: The base is always the small subscript number written after 'log.' In

\log_3 81

, the base is 3 and the argument is 81. Converting to exponential form (

3^? = 81

) makes the roles clear.

Mistake: Forgetting the default base. Students sometimes treat

\log 100

as a natural logarithm or assume an unwritten base of 2.

Correction: By convention,

\log

with no written base means base 10 (common logarithm), and

\ln

means base

e

. So

\log 100 = \log_{10} 100 = 2

Related Terms

Base in an Exponential Expression — Same base appears in the equivalent exponential form
Logarithm — The operation that uses a base
Logarithm Rules — Properties that depend on the base
Change of Base Formula — Converts between different logarithm bases
Common Logarithm — Logarithm with the specific base 10
Natural Logarithm — Logarithm with the specific base e