The exponential function

Exponential and logarithmic functions, grade 12 (4 units) Lesson 2 of 20Learning

The exponential function

In an exponential function the variable is in the exponent, and the base is a fixed number:

\displaystyle f(x)=a^{x}

\displaystyle a>0,\ a\neq1

A negative base has no value at some $\displaystyle x$ : $\displaystyle {(-4)^{\frac{1}{2}}}={\sqrt{-4}}$ is not a number.
The base 1 gives $\displaystyle {1^x}={1}$ for every $\displaystyle x$ – a straight line, not a new kind of function.

A table of

\displaystyle 2^x

and of

\displaystyle \left(\frac{1}{2}\right)^x

Each step of 1 in $\displaystyle x$ multiplies $\displaystyle 2^x$ by 2, and $\displaystyle \left(\frac{1}{2}\right)^x$ by $\displaystyle \frac{1}{2}$ :

$\displaystyle x$	$\displaystyle -2$	$\displaystyle -1$	$\displaystyle 0$	$\displaystyle 1$	$\displaystyle 2$	$\displaystyle 3$
$\displaystyle 2^x$	$\displaystyle \frac{1}{4}$	$\displaystyle \frac{1}{2}$	$\displaystyle 1$	$\displaystyle 2$	$\displaystyle 4$	$\displaystyle 8$
$\displaystyle \left(\frac{1}{2}\right)^x$	$\displaystyle 4$	$\displaystyle 2$	$\displaystyle 1$	$\displaystyle \frac{1}{2}$	$\displaystyle \frac{1}{4}$	$\displaystyle \frac{1}{8}$

\displaystyle f(x)=2^{x}

\displaystyle g(x)=\left(\frac{1}{2}\right)^{x}

\displaystyle y=0

The properties of

\displaystyle a^x

The graphs of $\displaystyle 2^x$ and of $\displaystyle \left(\frac{1}{2}\right)^x$ show the properties of every exponential function:

The domain: every $\displaystyle x$ .
The point on the $\displaystyle y$ -axis: $\displaystyle (0,1)$ , since $\displaystyle {a^0}={1}$ . There is no point on the $\displaystyle x$ -axis.
Positive for every $\displaystyle x$ : a positive base to any power is positive.
The horizontal asymptote $\displaystyle {y}={0}$ : when $\displaystyle a>1$ the graph comes close to it on the left, when $\displaystyle 0<a<1$ on the right.
When $\displaystyle a>1$ the function is increasing; when $\displaystyle 0<a<1$ it is decreasing.

Several bases

All the graphs of $\displaystyle a^x$ pass through $\displaystyle (0,1)$ ; the base decides how steep they are.

For $\displaystyle x>0$ the graph of the larger base is above: $\displaystyle 10^x>3^x>2^x$ .
For $\displaystyle x<0$ it is the other way round: $\displaystyle {10^{-1}}={\frac{1}{10}}$ is less than $\displaystyle {2^{-1}}={\frac{1}{2}}$ .

\displaystyle f(x)=2^{x}

\displaystyle g(x)=3^{x}

\displaystyle h(x)=10^{x}

\displaystyle \left(\frac{1}{a}\right)^x

, the mirror of

\displaystyle a^x

$\displaystyle {\left(\frac{1}{2}\right)^x}={\frac{1}{2^x}}={2^{-x}}$ : at $\displaystyle x$ it has the value $\displaystyle 2^x$ has at $\displaystyle -x$ .

So the graph of $\displaystyle \left(\frac{1}{2}\right)^x$ is the graph of $\displaystyle 2^x$ reflected in the $\displaystyle y$ -axis.
The same holds for every base: $\displaystyle {\left(\frac{1}{a}\right)^x}={a^{-x}}$ .

Exercise 1

The function: $\displaystyle {f(x)}={4^x}$ .

Calculate $\displaystyle f\left(-\frac{1}{2}\right)$ .

/ – 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

Calculate $\displaystyle f\left(\frac{3}{2}\right)$ .

/ – 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

The functions: $\displaystyle {f(x)}={3^x}$ , $\displaystyle {g(x)}={\left(\frac{1}{3}\right)^x}$ .

Mark the point where the graphs meet.

\displaystyle f(x)=3^{x}

\displaystyle g(x)=\left(\frac{1}{3}\right)^{x}

For which $\displaystyle x$ is $\displaystyle f(x)>g(x)$ ? Explain by the graphs.

Exercise 3

The graph of an exponential function $\displaystyle {f(x)}={a^x}$ passes through the point given.
Find $\displaystyle a$ , and write the function in a row of the board.

The point $\displaystyle A(2,9)$ .

\displaystyle A=(2,9)

The point $\displaystyle B(-2,16)$ .

\displaystyle B=(-2,16)

Exercise 4

The functions: $\displaystyle 5^x$ , $\displaystyle 0.3^x$ , $\displaystyle \left(\frac{3}{2}\right)^x$ , $\displaystyle \left(\frac{1}{2}\right)^{-x}$ .
Which of them are increasing, and which decreasing? Explain.

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

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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
$\displaystyle S=\int_{a}^{b}\left(f(x)-g(x)\right)dx$ $\displaystyle f(x)\geq g(x)$