What logarithms are and how to use them

Updated October 20, 2026

A logarithm answers the question: what power do I raise the base to, to get this number? log_b(x) = y means the same as b^y = x, so log₂ 8 = 3 because 2³ = 8. Logarithms undo exponents, which is why they are the tool for solving equations where the unknown is in the exponent.

What a logarithm is

Exponents and logarithms are two ways of writing the same fact. The statement 10³ = 1000 can be turned around to ask: 10 to what power is 1000? The answer, 3, is the logarithm of 1000 with base 10, written log₁₀ 1000 = 3.

In general, log_b(x) = y means b^y = x. The base b must be positive and not equal to 1, and x must be positive. Reading a log aloud as the question it asks makes many problems easy:

log₂ 32 = 5, because 2⁵ = 32.
log₅ 1 = 0, because any allowed base to the power 0 is 1.
log₇ 7 = 1, because 7¹ = 7.
log₃ (1/9) = −2, because 3⁻² = 1/9.
log₄ 2 = 1/2, because the square root of 4 is 2.

Common and natural logarithms

Two bases are used so often that they have their own buttons on a calculator. The common logarithm, written log with no base, uses base 10: log 100 = 2. The natural logarithm, written ln, uses base e, a constant about equal to 2.71828 that appears in growth, decay and calculus: ln e = 1.

Base 10 logs measure size in powers of ten, which is why they sit behind scales that cover a huge range of values. The pH of a solution is −log of its hydrogen ion concentration, and the decibel scale for sound intensity is also built on base 10 logarithms. Each step of 1 on a log scale is a tenfold change.

The rules of logarithms

Each rule comes from a rule of exponents. Because b^m × b^n = b^(m+n), multiplying numbers adds their logs, and the others follow the same way. All of them hold for any allowed base, as long as x and y are positive:

Product rule: log_b(xy) = log_b x + log_b y.
Quotient rule: log_b(x/y) = log_b x − log_b y.
Power rule: log_b(xⁿ) = n log_b x.
log_b 1 = 0 and log_b b = 1.
Inverse rules: log_b(bˣ) = x and b^(log_b x) = x. A log and an exponential with the same base undo each other.

ProblemSimplify $2 \log_{3} 6 - \log_{3} 4$ .

$2 \log_{3} 6$ $= \log_{3} 6^{2}$ $= \log_{3} 36$
Power rule: the 2 in front becomes an exponent.
$\log_{3} 36 - \log_{3} 4$ $= \log_{3} \frac{36}{4}$
Quotient rule: a difference of logs is the log of a quotient.
$\log_{3} 9$
36 divided by 4 is 9.
$\log_{3} 9$ $= 2$
3 to the power 2 is 9.

Answer $2$

Change of base

Calculators have keys for base 10 and base e only. To find a log in any other base, use the change of base formula: log_b x = ln x ÷ ln b, or equally log x ÷ log b. Either key gives the same answer, as long as you use the same one on top and bottom.

Estimate before you press any keys. Since 5² = 25 and 5³ = 125, log₅ 40 must lie between 2 and 3, a little closer to 2.

ProblemFind $\log_{5} 40$ to three decimal places.

$\log_{5} 40$ $= \frac{\ln 40}{\ln 5}$
Change of base, using natural logs.
$\approx \frac{3.6889}{1.6094}$
Evaluate each log on a calculator.
$\approx 2.292$
Divide. This fits the estimate between 2 and 3.
$5^{2.292}$ $\approx 40.0$
Check by raising 5 to the answer.

Answer $\approx 2.292$

Solving exponential equations

When the unknown is in an exponent, a logarithm brings it down. The steps are: isolate the power, take the log of both sides (or rewrite in log form), use the power rule, then divide.

Sometimes both sides can be written with the same base and no log is needed: 2ˣ = 32 means 2ˣ = 2⁵, so x = 5. When that is not possible, logs give the exact answer. For example, money growing at 5% a year doubles when 1.05ᵗ = 2, so t = ln 2 ÷ ln 1.05 ≈ 14.21 years.

Logarithmic equations go the other way: rewrite the log as a power. log₂(x + 3) = 5 means x + 3 = 2⁵ = 32, so x = 29. Always check that the value inside every log stays positive, because a solution that makes it zero or negative must be rejected.

ProblemSolve $5 \cdot 3^{x} = 70$ .

$3^{x}$ $= 14$
Isolate the power: divide both sides by 5 first, not after taking logs.
$\ln 3^{x}$ $= \ln 14$
Take the natural log of both sides.
$x \ln 3$ $= \ln 14$
Power rule: the exponent comes down in front.
$x$ $= \frac{\ln 14}{\ln 3}$ $\approx \frac{2.6391}{1.0986}$ $\approx 2.402$
Divide by ln 3. Check: 3² = 9 and 3³ = 27, so an answer between 2 and 3 makes sense.

Answer $x \approx 2.402$

Common mistakes

Log rules look like algebra rules, which tempts students into moves that are not allowed:

Splitting a log of a sum. log(x + y) is not log x + log y; the product rule is about log(xy).
Confusing a quotient of logs with a log of a quotient. log x ÷ log y is not log(x/y).
Mixing up (log x)² with log(x²). Only the second becomes 2 log x.
Taking the log of zero or a negative number. Neither is defined for real numbers.
Taking logs before isolating the power. In 5 × 3ˣ = 70, divide by 5 first.
Pressing log when the problem means ln, or the reverse. Check which base the question uses.

How to practise and check your work

Build fluency by converting back and forth between exponential and log form until it is automatic, then learn the rules by checking them with numbers: log 1000 = log 10 + log 100 is 3 = 1 + 2.

To check a solution, substitute it into the original equation; a calculator makes this quick for decimals. Estimating with nearby whole powers, as in the examples above, catches most errors in change of base. In Common Core high school math, students use logarithms to solve exponential equations and study logs as inverses of exponential functions in Algebra II and precalculus.

Questions

What is a logarithm in simple terms?

A logarithm is an exponent. log_b(x) asks what power you raise b to in order to get x, so log₁₀ 100 = 2 because 10² = 100.

What is the difference between log and ln?

log without a base usually means base 10, the common logarithm. ln means base e, about 2.71828, the natural logarithm. Both follow the same rules.

Can you take the log of a negative number?

Not with real numbers. A positive base raised to any real power is always positive, so the input to a log must be positive.

Why is the log of 1 equal to 0?

Because any allowed base raised to the power 0 equals 1. So log_b 1 = 0 for every base.

How do I calculate log base 2 on a calculator?

Use change of base: log₂ x = log x ÷ log 2, or ln x ÷ ln 2. For example, log₂ 10 ≈ 1 ÷ 0.30103 ≈ 3.322.