The laws of powers

Exponential and logarithmic functions, grade 12 (4 units) Lesson 1 of 20Learning

The laws of powers

In a power $\displaystyle a^x$ , $\displaystyle a$ is the base and $\displaystyle x$ the exponent. Powers with the same base are multiplied and divided through their exponents:

\displaystyle a^{x}\cdot a^{y}

\displaystyle =

\displaystyle a^{x+y}

\displaystyle \frac{a^{x}}{a^{y}}

\displaystyle =

\displaystyle a^{x-y}

\displaystyle (a^{x})^{y}

\displaystyle =

\displaystyle a^{x\cdot y}

A power of a product or of a quotient is the product or the quotient of the powers:

\displaystyle (a\cdot b)^{x}

\displaystyle =

\displaystyle a^{x}\cdot b^{x}

\displaystyle \left(\frac{a}{b}\right)^{x}

\displaystyle =

\displaystyle \frac{a^{x}}{b^{x}}

The exponents 0 and negative

The quotient law gives a meaning to the exponent 0 and to a negative exponent: $\displaystyle {\frac{a^{3}}{a^{3}}}={a^{0}}$ is 1, and $\displaystyle {\frac{a^{2}}{a^{5}}}={a^{-3}}$ is $\displaystyle \frac{1}{a^{3}}$ .

\displaystyle a^{0}

\displaystyle =

\displaystyle 1

\displaystyle a\neq0

\displaystyle a^{-x}

\displaystyle =

\displaystyle \frac{1}{a^{x}}

$\displaystyle {2^{-3}}={\frac{1}{2^{3}}}={\frac{1}{8}}$ .
$\displaystyle {\left(\frac{2}{3}\right)^{-2}}={\left(\frac{3}{2}\right)^{2}}={\frac{9}{4}}$ : a negative exponent turns the fraction over.

A rational exponent

A fraction in the exponent is a root: the denominator is the root’s index, the numerator the power. The base is positive.

\displaystyle a^{\frac{m}{n}}

\displaystyle =

\displaystyle \sqrt[n]{a^{m}}

\displaystyle a>0

$\displaystyle {8^{\frac{2}{3}}}={\left(\sqrt[3]{8}\right)^{2}}={2^{2}}={4}$ : it is easier to take the root first.
$\displaystyle {16^{-\frac{3}{4}}}={\frac{1}{16^{\frac{3}{4}}}}={\frac{1}{\left(\sqrt[4]{16}\right)^{3}}}={\frac{1}{8}}$ .
A root is a power: $\displaystyle {\sqrt{x}}={x^{\frac{1}{2}}}$ , $\displaystyle {\sqrt[3]{x^{2}}}={x^{\frac{2}{3}}}$ , $\displaystyle {\frac{1}{\sqrt{x}}}={x^{-\frac{1}{2}}}$ .
All the laws of powers hold for rational exponents too.

A worked example: a number

The example below calculates $\displaystyle 27^{\frac{2}{3}}\cdot9^{-\frac{1}{2}}$ by writing both bases as powers of 3:

27^{\frac{2}{3}}\cdot9^{-\frac{1}{2}}=\left(3^{3}\right)^{\frac{2}{3}}\cdot\left(3^{2}\right)^{-\frac{1}{2}}=3^{2}\cdot3^{-1}=3^{1}=3

A worked example: letters

The example below simplifies an expression with $\displaystyle a>0$ , and writes the result with a positive exponent:

\frac{(a^{3})^{2}\cdot a^{-5}}{\sqrt{a^{6}}}=\frac{a^{6}\cdot a^{-5}}{a^{3}}=\frac{a^{1}}{a^{3}}=a^{-2}=\frac{1}{a^{2}}

The root is a power: $\displaystyle {\sqrt{a^{6}}}={a^{\frac{6}{2}}}={a^{3}}$ .
The exponents add up in the numerator, $\displaystyle {6+(-5)}={1}$ , and subtract in the quotient, $\displaystyle {1-3}={-2}$ .

Exercise 1

Calculate.

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Exercise 2

In every expression $\displaystyle a>0$ and $\displaystyle x>0$ .
Simplify, and write the result with a positive exponent.

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Exercise 3

In every expression $\displaystyle a>0$ and $\displaystyle x>0$ .
Write the roots as powers, and simplify.

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Exercise 4

The value of each expression is the same for every $\displaystyle x$ .
Find it.

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Formulas: exponential and logarithmic functions, grade 12 (4 units)

Open as a page

Powers

A product of powers with the same base
$\displaystyle a^{x}\cdot a^{y}$ $\displaystyle =$ $\displaystyle a^{x+y}$
A quotient of powers with the same base
$\displaystyle \frac{a^{x}}{a^{y}}$ $\displaystyle =$ $\displaystyle a^{x-y}$
A power of a power
$\displaystyle (a^{x})^{y}$ $\displaystyle =$ $\displaystyle a^{x\cdot y}$
A power of a product
$\displaystyle (a\cdot b)^{x}$ $\displaystyle =$ $\displaystyle a^{x}\cdot b^{x}$
A power of a quotient
$\displaystyle \left(\frac{a}{b}\right)^{x}$ $\displaystyle =$ $\displaystyle \frac{a^{x}}{b^{x}}$
A negative exponent
$\displaystyle a^{-x}$ $\displaystyle =$ $\displaystyle \frac{1}{a^{x}}$
The exponent 0
$\displaystyle a^{0}$ $\displaystyle =$ $\displaystyle 1$ $\displaystyle a\neq0$
A rational exponent
$\displaystyle a^{\frac{m}{n}}$ $\displaystyle =$ $\displaystyle \sqrt[n]{a^{m}}$ $\displaystyle a>0$

Logarithms (within the domain)

The definition of the logarithm
$\displaystyle \log_{a}x=b\iff a^{b}=x$ $\displaystyle a>0,\ a\neq1,\ x>0$
$\displaystyle \log_{a}(a^{b})$ $\displaystyle =$ $\displaystyle b$
$\displaystyle a^{\log_{a}x}$ $\displaystyle =$ $\displaystyle x$
The logarithm of a product
$\displaystyle \log_{a}x+\log_{a}y$ $\displaystyle =$ $\displaystyle \log_{a}(x\cdot y)$
The logarithm of a quotient
$\displaystyle \log_{a}x-\log_{a}y$ $\displaystyle =$ $\displaystyle \log_{a}\left(\frac{x}{y}\right)$
The logarithm of a power
$\displaystyle \log_{a}(x^{b})$ $\displaystyle =$ $\displaystyle b\cdot\log_{a}x$
The natural logarithm – to the base $\displaystyle e$
$\displaystyle \ln x=\log_{e}x$

Growth and decay

The amount after $\displaystyle t$ units of time
$\displaystyle f(t)=f(0)\cdot q^{t}$
The growth / decay factor – $\displaystyle p$ the percent per unit of time
$\displaystyle q=1\pm\frac{p}{100}$
The half-life $\displaystyle T$
$\displaystyle q^{T}$ $\displaystyle =$ $\displaystyle \frac{1}{2}$

Derivatives

$\displaystyle (e^{x})^{\prime}$ $\displaystyle =$ $\displaystyle e^{x}$
$\displaystyle (\ln x)^{\prime}$ $\displaystyle =$ $\displaystyle \frac{1}{x}$
The derivative of $\displaystyle e^{f(x)}$
$\displaystyle \left[e^{f(x)}\right]^{\prime}$ $\displaystyle =$ $\displaystyle e^{f(x)}\cdot f^{\prime}(x)$
The derivative of $\displaystyle \ln(f(x))$
$\displaystyle \left[\ln(f(x))\right]^{\prime}$ $\displaystyle =$ $\displaystyle \frac{f^{\prime}(x)}{f(x)}$
The derivative of a product
$\displaystyle \left[f(x)\cdot g(x)\right]^{\prime}$ $\displaystyle =$ $\displaystyle f^{\prime}(x)\cdot g(x)+f(x)\cdot g^{\prime}(x)$
The derivative of a quotient
$\displaystyle \left[\frac{f(x)}{g(x)}\right]^{\prime}$ $\displaystyle =$ $\displaystyle \frac{f^{\prime}(x)\cdot g(x)-f(x)\cdot g^{\prime}(x)}{\left[g(x)\right]^{2}}$
The derivative of a composite function
$\displaystyle \left[f(u(x))\right]^{\prime}$ $\displaystyle =$ $\displaystyle f^{\prime}(u)\cdot u^{\prime}(x)$
The tangent’s equation at the point where $\displaystyle {x}={a}$
$\displaystyle y-f(a)=f^{\prime}(a)(x-a)$

Integrals and the area

$\displaystyle \int e^{x}\,dx=e^{x}+C$
$\displaystyle \int e^{mx+b}\,dx=\frac{e^{mx+b}}{m}+C$ $\displaystyle m\neq0$
In any interval where $\displaystyle x\neq0$
$\displaystyle \int\frac{1}{x}\,dx=\ln|x|+C$
$\displaystyle \int\frac{1}{mx+b}\,dx=\frac{\ln|mx+b|}{m}+C$ $\displaystyle m\neq0$
$\displaystyle F$ an antiderivative of $\displaystyle f$
$\displaystyle \int_{a}^{b}f(x)\,dx=F(b)-F(a)$
The area between two graphs
$\displaystyle S=\int_{a}^{b}\left(f(x)-g(x)\right)dx$ $\displaystyle f(x)\geq g(x)$