01 Foundations

Exponents and Logarithms

Understand powers, roots, exponent rules, logarithms, and natural logarithms.

mathematics · exponents

Exponents

An exponent tells us how many times a number or variable is multiplied by itself.

For example:

x^3=x\cdot x\cdot x

2.1 Multiplying powers with the same base

When multiplying powers with the same base, add the exponents:

x^2\cdot x^3=x^{2+3}=x^5

2.2 Dividing powers with the same base

When dividing powers with the same base, subtract the exponents:

\frac{x^5}{x^2}=x^{5-2}=x^3

Here, $x\ne0$ .

2.3 A power raised to another power

When a power is raised to another power, multiply the exponents:

(x^2)^3=x^{2\cdot3}=x^6

2.4 Zero exponent

Any non-zero number raised to the power zero equals 1:

x^0=1,\qquad x\ne0

2.5 Negative exponents

A negative exponent represents a reciprocal:

x^{-3}=\frac{1}{x^3},\qquad x\ne0

2.6 Fractional exponents

A fractional exponent represents a root. For example:

x^{1/2}=\sqrt{x}

More generally:

x^{1/n}=\sqrt[n]{x}

2.7 Combining exponent rules

For example:

\frac{x^6}{x^2\cdot x} = \frac{x^6}{x^3} = x^3

The rules for exponents are useful for simplifying expressions and appear throughout algebra, calculus, and machine learning.

Logarithms

A logarithm is the inverse operation of exponentiation.

If:

\[b^c=a\]

then:

\log_b(a)=c

In words, $\log_b(a)$ asks: To what power must the base $$b$$ be raised to produce $$a$$ ?

For example:

\[2^4=16\]

Therefore:

\log_2(16)=4

3.1 Basic logarithm examples

\log_3(27)=3

because $$3^3=27$$ .

\log_5(1)=0

because $$5^0=1$$ .

\log_{10}(100)=2

because $$10^2=100$$ .

A logarithm can also produce a fractional or negative result:

\log_2(\sqrt{2})=\frac12

because $2^{1/2}=\sqrt{2}$ .

\log_2\left(\frac12\right)=-1

because $2^{-1}=\frac12$ .

3.2 Logarithm rules

Product rule

The logarithm of a product is the sum of the logarithms:

\log_b(ac)=\log_b(a)+\log_b(c)

Quotient rule

The logarithm of a quotient is the difference of the logarithms:

\log_b\left(\frac{a}{c}\right) = \log_b(a)-\log_b(c)

Power rule

The exponent can be brought in front as a multiplier:

\log_b(a^n)=n\log_b(a)

Logarithms do not distribute over addition:

\log_b(a+c)\ne\log_b(a)+\log_b(c)

3.3 Natural logarithms

The natural logarithm is a logarithm with base $$e$$ , where:

e\approx2.71828

It is written as:

\ln(x)=\log_e(x)

For example:

\ln(e^2)=2

and:

\ln\left(\frac{1}{e^3}\right)=-3

If:

\ln(x)=2

then exponentiating both sides gives:

\[x=e^2\]

Similarly, if:

\log_2(x)=4

then:

\[x=2^4=16\]

The natural logarithm appears frequently in calculus, probability, statistics, and machine learning.