Powers and logarithms

A quick reference ranging from core operational rules to the details that prevent domain and algebraic-manipulation errors.

Basic notation and definitions

Power

$a^{n}$ means multiplying $a$ by itself $n$ times, when $n$ is a natural number.

Base and exponent

The base is the number being raised; the exponent tells how many times the factor is repeated.

Logarithm

$\log_{b} (x) = y$ is equivalent to saying that $b^{y}$ equals $x$ .

Fundamental properties of powers

Property	Formula	Conditions	Example
Product	$a^{m} \cdot a^{n} = a^{m + n}$	Same base	$2^{3} \cdot 2^{4} = 2^{7}$
Quotient	$\frac{a^{m}}{a^{n}} = a^{m - n}$	$a \neq 0$	$\frac{5^{6}}{5^{2}} = 5^{4}$
Power of a power	$(^{a^{m}) n} = a^{m n}$	Valid in the usual domains	$(^{3^{2}) 4} = 3^{8}$
Power of a product	$(^{a b) n} = a^{n} b^{n}$	Product as the base	$(^{2 \cdot 3) 2} = 2^{2} \cdot 3^{2}$
Power of a quotient	$(^{\frac{a}{b}) n} = \frac{a^{n}}{b^{n}}$	$b \neq 0$	$(^{\frac{6}{2}) 3} = \frac{6^{3}}{2^{3}}$
Power of 1	$1^{n} = 1$	For every exponent	$1^{100} = 1$
Power of 0	$0^{n} = 0$	$n > 0$	$0^{5} = 0$
Zero exponent	$a^{0} = 1$	$a \neq 0$	$7^{0} = 1$

Warning: 0^0 is an indeterminate form in the most common context and must be handled with care.

Negative and fractional exponents and radicals

a^{- n} = \frac{1}{a^{n}}

$a^{- n}$ requires $a \neq 0$ .
$a^{\frac{1}{n}}$ over the reals requires attention to the sign of $a$ and to whether $n$ is even or odd.
For $a < 0$ , even-index roots are not real; an odd-index root does have a real value.
Rational exponents let us rewrite many radicals as powers and vice versa.

Definition of a logarithm

Definition

$\log_{b} (x) = y$ if and only if $b^{y}$ equals $x$ .

Existence conditions

$b > 0, b \neq 1, x > 0$ .

Interpretation

A logarithm gives the exponent to which the base must be raised to obtain the argument.

Logarithms invert exponentiation, which makes them useful when moving between multiplicative and exponential growth.

Properties of logarithms

Property	Formula	Conditions	Example
Product	$\log_{b} (x y) = \log_{b} (x) + \log_{b} (y)$	$x, y > 0$	$\log_{2} (8 \cdot 4) = \log_{2} (8) + \log_{2} (4)$
Quotient	$\log_{b} (\frac{x}{y}) = \log_{b} (x) - \log_{b} (y)$	$x, y > 0$	$\log_{10} (\frac{100}{10}) = 2 - 1$
Power	$\log_{b} (x^{k}) = k \cdot \log_{b} (x)$	$x > 0$	$\log_{3} (27) = \log_{3} (3^{3}) = 3$
Base and argument	$\log_{b} (b) = 1$	$b > 0, b \neq 1$	$\log_{5} (5) = 1$
One	$\log_{b} (1) = 0$	$b > 0, b \neq 1$	$\log_{7} (1) = 0$
Inverse	$\log_{b} (b^{x}) = x$	$b > 0, b \neq 1$	$\log_{2} (2^{8}) = 8$

Change of base and common logarithms

Change of base

\log_{a} (b) = \frac{\log_{c} (b)}{\log_{c} (a)}

With $c$ any valid base. This is the most convenient form to remember.

Natural logarithm

$\ln (x)$ is the logarithm to base $e$ , widely used in calculus and continuous growth.

Common logarithm

$\log (x)$ or $\log_{10} (x)$ often denotes the base-10 logarithm, especially in school mathematics.

\log_{2} (8) = \frac{\ln (8)}{\ln (2)} = 3

Exercises with hidden solutions

Each exercise covers a typical case; open the solution to reveal the answer.

1. Product of powers with the same base

Simplify $2^{3} \cdot 2^{5}$ .

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Answer: $2^{8}$ = 256

Add the exponents because the base is the same.

2. Quotient of powers with the same base

Simplify $\frac{5^{7}}{5^{2}}$ .

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Answer: $5^{5}$ = 3125

Subtract the exponents: $7 - 2 = 5$ .

3. Power of a power

Simplify $(^{3^{2}) 4}$ .

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Answer: $3^{8}$ = 6561

Multiply the exponents: $2 \cdot 4 = 8$ .

4. Power of a product

Simplify $(^{2 \cdot 4) 3}$ and rewrite it by separating the bases.

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Answer: $8^{3}$ = $2^{3}$ · $4^{3}$ = 512

Distribute the exponent over each factor.

5. Zero exponent

Calculate $9^{0}$ .

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Answer: $1$

Every nonzero base raised to zero equals 1.

6. Negative exponent

Calculate $4^{- 2}$ .

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Answer: $\frac{1}{16}$

A negative exponent takes the reciprocal: $4^{- 2} = \frac{1}{4^{2}}$ .

7. Fractional exponent

Calculate $16^{\frac{1}{2}}$ .

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Answer: $4$

$\frac{1}{2}$ denotes the square root.

8. General rational exponent

Calculate $27^{\frac{2}{3}}$ .

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Answer: $9$

First take the cube root: $27^{\frac{1}{3}} = 3$ , then square the result.

9. Logarithm from the definition

Find $\log_{2} (32)$ .

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Answer: $5$

Because $2^{5} = 32$ .

10. Sum of logarithms

Simplify $\log_{10} (2) + \log_{10} (5)$ .

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Answer: $\log_{10} (10) = 1$

Use the product property: $\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

11. Difference of logarithms

Simplify $\log_{3} (81) - \log_{3} (3)$ .

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Answer: $3$

$\log_{3} (81) = 4$ and $\log_{3} (3) = 1$ , so $4 - 1 = 3$ .

12. Change of base

Rewrite $\log_{2} (8)$ using natural logarithms.

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Answer: $\frac{\ln (8)}{\ln (2)} = 3$

Apply the formula $\frac{\log_{b} (x)}{\ln (b)}$ .

13. Domain check

Decide whether $\log_{5} (- 2)$ is defined over the reals.

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Answer: it is not defined over the reals.

The argument of a real logarithm must be positive.

Useful notes and cases to remember

A logarithm’s argument must always be positive: $x > 0$ .
A logarithm’s base must satisfy $b > 0, b \neq 1$ .
Do not apply logarithm properties mechanically before checking that the expression lies in the correct domain.
Power rules require care with a zero base, negative bases and non-integer exponents.
A sum inside a logarithm cannot be split: $\log (x + y) \neq \log (x) + \log (y)$ .
Likewise, ${(x + y)}^{n} \neq x^{n} + y^{n}$ in general.

Practical rule: check the domain first, then apply algebraic properties, and simplify numerical values only at the end.

This page is a compact but complete reference for review, exercises and quick formula checks.