The logarithm log_a b is the exponent the base a has to be raised to in order to give the number b. Writing log₂ 8 = 3 means precisely that 2³ = 8. So a logarithm is the inverse of raising to a power, the way subtraction is the inverse of addition.

Why cannot the base of a logarithm be 1?

Because one raised to any power gives 1. The equation 1^c = 8 has no solution and the equation 1^c = 1 has infinitely many, so a logarithm to base 1 would either not exist or not be unique. For the same reason the base has to be positive.

Why is there no logarithm of a negative number?

Because a power of a positive number is always positive: no exponent c satisfies 2^c = −8. The domain of a logarithm is therefore the positive numbers only, and log(−8) or log 0 has no meaning.

What is the difference between log and ln?

They are the same logarithm to two different bases. The notation log x means the common logarithm, to base 10, and shows up in measurement scales — decibels, pH, the Richter scale. The notation ln x means the natural logarithm to base e ≈ 2.718 and is the working tool of calculus.

How do you compute a logarithm without a calculator?

By writing the argument as a power of the base. For example log₃ 81 works out because 81 = 3⁴, so the logarithm is 4. When the number is not an exact power, the result is irrational and is given as an approximation — which is why textbook problems are chosen so that the power is exact.

Advanced level

The logarithm as an inverse operation

A logarithm answers the question of which power a base has to be raised to in order to give a number. Learn the definition of log_a b, the conditions on the base and the argument, the common and natural logarithms, and the four properties that turn multiplication into addition.

Before you start

This topic builds on earlier ideas. Before you start, it's worth working through the lessons below — they'll make everything click:

PowersA power is shorthand for multiplying the same factor by itself. Learn the base and the exponent, the laws of exponents, powers of a product and of a quotient, zero and negative exponents, the monotonicity of exponentiation, and scientific notation.
RootsA root is the inverse of raising to a power. Learn square and cube roots, arithmetic with roots, taking a factor out of a radical and absorbing one under it, estimating and comparing roots, and rationalising a denominator.

Where this is used

Real situations where you count exactly the way this lesson teaches:

Noise and hearing
A conversation runs at about 60 dB and a busy street at 90 dB. The 30 dB gap does not mean half again as loud — it means 10³ = 1000 times the sound power, because the decibel is a logarithmic unit rather than an ordinary scale.
Convert it here: Decibels
Saving money
A deposit pays 5% a year, and the question is after how many years the capital doubles. Solving 1.05ⁿ = 2 gives n = log 2 : log 1.05 ≈ 14.2 years. Without a logarithm all that is left is multiplying year by year and guessing.
Chemistry laboratory
Acidity is reported as pH = −log[H⁺]. Juice at pH 3 has a hydrogen-ion concentration 100 times higher than tap water at pH 5, although only 2 separates them on the scale — every pH unit is a factor of ten.
Software development
A binary search through a sorted set of a million records needs log₂ 1,000,000 ≈ 20 comparisons, while scanning one by one needs 500,000 on average. That difference decides whether a database answers instantly or after a second.

All formulas

Definition of a logarithm
$\log_a b = c \iff a^c = b \quad (a > 0,\ a \neq 1,\ b > 0)$
a logarithm is the exponent the base is raised to
Two logarithms straight from the definition
$\log_a 1 = 0 \qquad \log_a a = 1$
because a⁰ = 1 and a¹ = a
Logarithm of a product
$\log_a (x \cdot y) = \log_a x + \log_a y$
multiplication turns into addition
Logarithm of a quotient
$\log_a \frac{x}{y} = \log_a x - \log_a y$
division turns into subtraction
Logarithm of a power
$\log_a x^k = k \cdot \log_a x$
the exponent comes out in front
Inverse operations
$a^{\log_a x} = x \qquad \log_a a^k = k$
raising to a power and taking a logarithm cancel each other
Common and natural logarithm
$\log x = \log_{10} x \qquad \ln x = \log_{e} x$
two bases used so often that they have their own notation

The statement $2^3 = 8$ can be read three ways, and each one leaves a different unknown. When the result is unknown, we compute a power. When the base is unknown, we take a root: $\sqrt[3]{8} = 2$ . That leaves a third possibility — the exponent is unknown:

2^x = 8

The answer is $3$ , and the operation that produces it is called a logarithm. It is the missing link beside powers and roots: a power asks for the result, a root asks for the base, and a logarithm asks for the exponent.

Definition

\log_a b = c \iff a^c = b \qquad (a > 0,\ a \neq 1,\ b > 0)

Read: the logarithm of $b$ to base $a$ is the exponent that $a$ must be raised to in order to give $b$ . The number $b$ is called the argument.

The easiest check is to rewrite the logarithm as a power:

\log_2 8 = 3, \ \text{because} \ 2^3 = 8 \qquad \log_{10} 1000 = 3, \ \text{because} \ 10^3 = 1000

\log_5 25 = 2 \qquad \log_3 81 = 4 \qquad \log_7 1 = 0

The three conditions in brackets are not decoration — each follows from how powers behave:

$a > 0$ and $a \neq 1$ — one raised to any power gives $1$ , so the equation $1^c = 8$ would have no solution while $1^c = 1$ would have infinitely many;
$b > 0$ — a power of a positive number is always positive, so no exponent gives $2^c = -8$ . The logarithm of a negative number and of zero does not exist.

Two values follow straight from the definition and are worth knowing by heart:

\log_a 1 = 0 \qquad \log_a a = 1

because $a^0 = 1$ and $a^1 = a$ .

Compute log₂ 64, log₁₀ 0.01 and log₄ 2.

Common and natural logarithms

Two bases turn up so often that they have a shorthand:

\log x = \log_{10} x \qquad \ln x = \log_{e} x

The common logarithm (base $10$ ) sits behind every scale on which the next step means a factor of ten: decibels, pH, the Richter scale. The natural logarithm has base $e \approx 2.718$ — the number that appears wherever something grows or decays continuously.

Properties

Every property of the logarithm is a law of exponents rewritten. Take $x = a^p$ and $y = a^q$ , that is $\log_a x = p$ and $\log_a y = q$ . Then

x \cdot y = a^p \cdot a^q = a^{p+q},

which says exactly that the logarithm of a product is the sum of the logarithms:

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

In the same way dividing powers gives subtraction, and raising a power to a power gives multiplication:

\log_a \frac{x}{y} = \log_a x - \log_a y \qquad \log_a x^k = k \cdot \log_a x

That is the whole story of the logarithm: it turns multiplication into addition and exponentiation into multiplication. Before calculators existed, this is precisely how many-digit numbers were multiplied — by reading logarithms off a table and adding them.

The drawings show the same thing. The numbers $1, 2, 4, 8, 16$ run apart faster and faster:

Each number is twice the previous one, so the gaps between them double as well.

Their logarithms to base $2$ sit at equal spacings:

Multiplying by 2 on the left corresponds to adding 1 on the right — which is what the product rule says.

The last pair of formulas states outright that taking a logarithm and raising to a power cancel each other:

a^{\log_a x} = x \qquad \log_a a^k = k

Compute log₃ 18 + log₃ 4.5 and log₂ 40 − log₂ 5.

What it is for

A logarithm is the answer wherever the unknown sits in the exponent. A deposit paying $5\%$ multiplies the capital by $1.05$ each year, so the question of how many years it takes to double is the equation

1.05^n = 2 \quad\Rightarrow\quad n = \frac{\log 2}{\log 1.05} \approx 14.2

The same construction describes radioactive decay, loan repayment and the running time of an algorithm. Logarithms also sit behind every scale on which a step means a factor rather than an increment: 30 decibels of difference is a thousandfold in sound power, and two pH units are a hundredfold in concentration. What that comes to in bels and nepers is what the decibel calculator works out.

Practice

Work through a set of exercises — they get harder as you go. At the end you'll see your score and the mistakes worth reviewing.

Exercise 1 of 8Score: 0

log₁₀ 100 =

Common mistakes

A logarithm of a sum — $\log_a (x + y)$ is not $\log_a x + \log_a y$ . The rules cover products and quotients, not addition.
Splitting a logarithm into a quotient of logarithms — $\log_a \tfrac{x}{y}$ is $\log_a x - \log_a y$ , not $\tfrac{\log_a x}{\log_a y}$ .
A logarithm of a negative number or of zero — it does not exist; the domain of a logarithm is the positive numbers.
Swapping the base with the argument — $\log_2 8 = 3$ , but $\log_8 2 = \tfrac{1}{3}$ . The base is the one written below.
Treating a negative result as an error — $\log_{10} 0.01 = -2$ is correct: the result may be negative even though the argument may not.

Formula card

Topic: Logarithms

Definition of a logarithm
$\log_a b = c \iff a^c = b \quad (a > 0,\ a \neq 1,\ b > 0)$
a logarithm is the exponent the base is raised to
Two logarithms straight from the definition
$\log_a 1 = 0 \qquad \log_a a = 1$
because a⁰ = 1 and a¹ = a
Logarithm of a product
$\log_a (x \cdot y) = \log_a x + \log_a y$
multiplication turns into addition
Logarithm of a quotient
$\log_a \frac{x}{y} = \log_a x - \log_a y$
division turns into subtraction
Logarithm of a power
$\log_a x^k = k \cdot \log_a x$
the exponent comes out in front
Inverse operations
$a^{\log_a x} = x \qquad \log_a a^k = k$
raising to a power and taking a logarithm cancel each other
Common and natural logarithm
$\log x = \log_{10} x \qquad \ln x = \log_{e} x$
two bases used so often that they have their own notation

The numbers 1, 2, 4, 8, 16 on an ordinary axis — each one twice as far out as the one before.

The logarithms of those same numbers to base 2 sit at equal spacings — multiplying by 2 became adding 1.

Frequently asked questions

$A fraction bar and a percent sign rendered as geometric forms, with a circle split into equal orbital slices, against a cosmic nebula in violet and magenta$

Numbers

A percentage is a fraction in disguise — three ways to write one number

1/2, 0.5 and 50% are not three different things but three ways of writing the same number. Here is where the fraction bar, the decimal point and the % sign came from, why 1/3 never ends, why +20% and −20% do not return you to the start, and why fractions are the first serious wall in school mathematics.

11 min