AOS1 Topic 8: Exponential and Logarithmic Functions

The function $f(x)=a^x$, where $a\in {\mathbb{R}}^{+}\setminus \{1\}$, is an exponential function. The shape of the graph depends on whether $a>1$ or $0<a<1$.

Key values are $f(-1)=\frac{1}{a}$, $f(0)=1$ and $f(1)=a$.

The maximal domain is $\mathbb{R}$ and the range is ${\mathbb{R}}^{+}$.

The x-axis is a horizontal asymptote.

An exponential function with $a>1$ is strictly increasing, and an exponential function with $0<a<1$ is strictly decreasing. In both cases, the function is one-to-one.

Graphing Transformations of $f(x)=a^x$

Translations

If the translation $(x,y)\to (x+h,y+k)$ is applied to the graph of $y=a^x$, then the image has equation $y=a^{x-h}+k$.

The horizontal asymptote of the image has equation $y=k$.

The range of the image is $(k,\mathrm{\infty })$.

Example:

Sketch the graph and state the range of $y=2^{x-1}+2$

Solution:

Explanation

The graph of $y=2^x$ is translated 1 unit in the positive direction of the x-axis and 2 units in the positive direction of the y-axis.

The mapping is $(x,y)\to (x+1,y+2)$.

Translation of key points:

• $(-1,\frac{1}{2})\to (0,\frac{5}{2})$
• $(0,1)\to (1,3)$
• $(1,2)\to (2,4)$

The range of the function is $(2,\mathrm{\infty })$.

Reflections

If a reflection in the x-axis, given by the mapping (x, y) → (x, −y), is applied to the graph of $y=a^x$, then the image has equation $y=-a^x$.

• The horizontal asymptote of the image has equation $y=0$.
• The range of the image is $(-\mathrm{\infty },0)$.

If a reflection in the y-axis, given by the mapping (x, y) → (−x, y), is applied to the graph of $y=a^x$, then the image has equation $y=a^{-x}$. This can also be written as $y=\frac{1}{a^x}$ or $y={\left(\frac{1}{a}\right)}^x$.

• The horizontal asymptote of the image has equation $y=0$.
• The range of the image is $(0,\mathrm{\infty })$.

Example:

Sketch the graph of $y=-3^x$

Solution

Explanation

Reflection of $y=3^x$ in the x-axis

The graph of $y=3^x$ is reflected in the x-axis using the mapping $(x,y)\to (x,-y)$.

Reflection of key points:

• $(-1,\frac{1}{3})\to (-1,-\frac{1}{3})$
• $(0,1)\to (0,-1)$
• $(1,3)\to (1,-3)$

Dilations

For $k>0$, if a dilation of factor $k$ from the x-axis, given by the mapping $(x,y)\to (x,ky)$, is applied to the graph of $y=a^x$, then the image has equation $y=kax$.

• The horizontal asymptote of the image has equation $y=0$.
• The range of the image is $(0,\mathrm{\infty })$.

For $k>0$, if a dilation of factor $k$ from the y-axis, given by the mapping $(x,y)\to (kx,y)$, is applied to the graph of $y=a^x$, then the image has equation $y=a^{x/k}$.

• The horizontal asymptote of the image has equation $y=0$.
• The range of the image is $(0,\mathrm{\infty })$.

Example:

Sketch the graphs of the following:

a) $y=3\times 5^x$

b) $y=0.2\times 8^x$

Solution:

Explanation:

(a)

The graph of $y=5^x$ is dilated by factor 3 from the x-axis.

The mapping is $(x,y)\to (x,3y)$.

Dilation of Key Points

• (−1, 1/5) → (−1, 3/5)
• (0, 1) → (0, 3)
• (1, 5) → (1, 15)

(b)

The graph of $y=8^x$ is dilated by a factor of $\frac{1}{5}$ from the x-axis.

The mapping is $(x,y)\to (x,\frac{1}{5}y)$.

Dilation of Key Points:

• $(-1,\frac{1}{8})\to (-1,\frac{1}{40})$
• $(0,1)\to (0,\frac{1}{5})$
• $(1,8)\to (1,\frac{8}{5})$

Note:

Since $\frac{9^x}{2}={\left(9^{\frac{1}{2}}\right)}^x=3^x$, the graph of $y=\frac{9^x}{2}$ is the same as the graph of $y=3^x$.Similarly, the graph of $y=3^{2x}$ is the same as the graph of $y=9^x$.

A translation parallel to the x-axis results in a dilation from the x-axis. For example, if the graph of $y=5^x$ is translated 3 units in the positive direction of the x-axis, then the image is the graph of $y=5^{x-3}$, which can be written as $y=5^{-3}\times 5^x$. Hence, a translation of 3 units in the positive direction of the x-axis is equivalent to a dilation of factor $5^{-3}$ from the x-axis.

Combinations of Transformations

Combinations of transformations refer to the application of multiple transformations to a function or a geometric object. These transformations can include translations, reflections, dilations, and rotations, among others. By combining these transformations, you can create more complex changes to the object's position, size, and orientation. For example, you can translate a shape, then reflect it across a line, and finally dilate it with respect to a point. These combinations allow for a wide range of modifications to the original object.

The Exponential Function $f(x)=e^x$

In the previous section, we explored the family of exponential functions $f(x)=a^x$, where $a\in {\mathbb{R}}^{+}\setminus \{1\}$. One particular member of this family is of great importance in mathematics. This function has the rule $f(x)=e^x$, where $e$ is Euler’s number, named after the eighteenth-century Swiss mathematician Leonhard Euler.

Euler’s number is defined as follows:

Euler’s Number

$$e=\underset{n\to \mathrm{\infty }}{lim}{(1+\frac{1}{n})}^n$$

To see what the value of $e$ might be, we could try large values of $n$ and use a calculator to evaluate ${(1+\frac{1}{n})}^n$, as shown in the table on the right. As $n$ is taken larger and larger, it can be seen that ${(1+\frac{1}{n})}^n$ approaches a limiting value (≈ 2.71828).

$n$	${(1+\frac{1}{n})}^n$
100	$(1.01{)}^{100}=2.704813\dots$
1,000	$(1.001{)}^{1000}=2.716923\dots$
10,000	$(1.0001{)}^{10000}=2.718145\dots$
100,000	$(1.00001{)}^{100000}=2.718268\dots$
1,000,000	$(1.000001{)}^{1000000}=2.718280\dots$

Like $\pi$, the number $e$ is irrational: $e=2.718281828459045\dots$

The function $f(x)=e^x$ is very important in mathematics.

Exponential Equations

One method for solving exponential equations is to use the one-to-one property of exponential functions:

$$a^x=a^y⟹x=y,\text{ for }a\in {\mathbb{R}}^{+}\setminus \{1\}$$

Index Laws

For all positive numbers $a$ and $b$ and all real numbers $x$ and $y$:

$$a^x\times a^y=a^{x+y}$$ $$\frac{a^x}{a^y}=a^{x-y}$$ $$(a^x{)}^y=a^{xy}$$ $$(ab{)}^x=a^x{b}^x$$ $${\left(\frac{a}{b}\right)}^x=\frac{a^x}{b^x}$$ $$a^{-x}=\frac{1}{a^x}$$ $$\frac{1}{a^x}=a^{-x}$$ $$a^0=1$$

Logarithms

Consider the statement $2^3=8$. This may be written in an alternative form: ${\log }_{2}8=3$, which is read as ‘the logarithm of 8 to the base 2 is equal to 3’.

For $a\in {\mathbb{R}}^{+}\setminus \{1\}$, the logarithm function with base $a$ is defined as follows:

$$a^x=y\text{ is equivalent to }{\log }_{a}y=x$$

Note: Since $a^x$ is positive, the expression ${\log }_{a}y$ is only defined when $y$ is positive.

Further examples:

• $3^2=9$ is equivalent to ${\log }_{3}9=2$
• ${10}^4=\mathrm{10,000}$ is equivalent to ${\log }_{10}\mathrm{10,000}=4$
• $a^0=1$ is equivalent to ${\log }_{a}1=0$

Inverse Functions

For each base $a\in {\mathbb{R}}^{+}\setminus \{1\}$, the exponential function $f:\mathbb{R}\to \mathbb{R},f(x)=a^x$ is one-to-one and so has an inverse function.

Let $a\in {\mathbb{R}}^{+}\setminus \{1\}$. The inverse of the exponential function $f:\mathbb{R}\to \mathbb{R},f(x)=a^x$ is the logarithmic function $f^{-1}:{\mathbb{R}}^{+}\to \mathbb{R},f^{-1}(x)={\log }_{a}x$.

$${\log }_a(a^x)=x\text{ for all }x\in \mathbb{R}$$ $$a^{{\log }_{a}x}=x\text{ for all }x\in {\mathbb{R}}^{+}$$

Because they are inverse functions, the graphs of $y=a^x$ and $y={\log }_{a}x$ are reflections of each other in the line $y=x$.

The Natural Logarithm

Earlier in the chapter we defined the number $e$ and the important function $f(x)=e^x$. The inverse of this function is $f^{-1}(x)={\log }_{e}x$. Because the logarithm function with base $e$ is known as the natural logarithm, the expression ${\log }_{e}x$ is also written as $\ln x$.

The Common Logarithm

The function $f(x)={\log }_{10}x$ has both historical and practical importance. When logarithms were used as a calculating device, it was often base 10 that was used. By simplifying calculations, logarithms contributed to the advancement of science, and especially of astronomy. In schools, books of tables of logarithms were provided for calculations and this was done up to the 1970s.

Base 10 logarithms are used for scales in science such as the Richter scale, decibels, and pH. You can understand the practicality of base 10 by observing:

$${\log }_{10}10=1$$ $${\log }_{10}100=2$$ $${\log }_{10}1000=3$$ $${\log }_{10}\mathrm{10,000}=4$$ $${\log }_{10}0.1=-1$$ $${\log }_{10}0.01=-2$$ $${\log }_{10}0.001=-3$$ $${\log }_{10}0.0001=-4$$

Laws of Logarithms

The index laws are used to establish rules for computations with logarithms.

Law 1: Logarithm of a Product

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

$${\log }_a(mn)={\log }_{a}m+{\log }_{a}n$$

Proof

Let ${\log }_{a}m=x$ and ${\log }_{a}n=y$, where $m$ and $n$ are positive real numbers. Then $a^x=m$ and $a^y=n$, and therefore

$$mn=a^x\times a^y=a^{x+y}$$ (using the first index law)

Hence

$${\log }_a(mn)=x+y={\log }_{a}m+{\log }_{a}n$$

For example:

$${\log }_{10}200+{\log }_{10}5={\log }_{10}(200\times 5)={\log }_{10}1000=3$$

Law 2: Logarithm of a Quotient

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

$${\log }_a\left(\frac{m}{n}\right)={\log }_{a}m-{\log }_{a}n$$

Proof

Let ${\log }_{a}m=x$ and ${\log }_{a}n=y$, where $m$ and $n$ are positive real numbers. Then as before, $a^x=m$ and $a^y=n$, and therefore

$$\frac{m}{n}=\frac{a^x}{a^y}=a^{x-y}$$ (using the second index law)

Hence

$${\log }_a\left(\frac{m}{n}\right)=x-y={\log }_{a}m-{\log }_{a}n$$

For example:

$${\log }_{2}32-{\log }_{2}8={\log }_2\left(\frac{32}{8}\right)={\log }_{2}4=2$$

Law 3: Logarithm of a Power

${\log }_a(m^p)=p{\log }_{a}m$

Proof

Let ${\log }_{a}m=x$. Then $a^x=m$, and therefore

$$m^p=(a^x{)}^p=a^{xp}$$ (using the third index law)

Hence

$${\log }_a(m^p)=xp=p{\log }_{a}m$$

For example:

$${\log }_{2}32={\log }_2(2^5)=5$$

Law 4: Logarithm of $\frac{1}{m}$

${\log }_a(m^{-1})=-{\log }_{a}m$

Proof

Use logarithm law 3 with $p=-1$.

For example:

$${\log }_a\left(\frac{1}{2}\right)={\log }_a(2^{-1})=-{\log }_{a}2$$

Law 5: Special Values

${\log }_{a}1=0$ and ${\log }_{a}a=1$

Proof

Since $a^0=1$, we have ${\log }_{a}1=0$.

Since $a^1=a$, we have ${\log }_{a}a=1$.

Graphing Logarithmic Functions

The graphs of $y=e^x$ and its inverse function $y={\log }_{e}x$ are shown on the one set of axes.

For each base $a\in {\mathbb{R}}^{+}\setminus \{1\}$, the graph of the logarithmic function $f(x)={\log }_{a}x$ has the following features:

• Key values are $f\left(\frac{1}{a}\right)=-1$, $f(1)=0$, and $f(a)=1$.
• The maximal domain is ${\mathbb{R}}^{+}$ and the range is $\mathbb{R}$.
• The y-axis is a vertical asymptote.

A logarithmic function with $a>1$ is strictly increasing, and a logarithmic function with $0<a<1$ is strictly decreasing. In both cases, the function is one-to-one.

Graphing Transformations of $f(x)={\log }_{a}x$

We now look at transformations applied to the graph of $f(x)={\log }_{a}x$ where $a>1$. We make the following general observations:

• The graph of $y={\log }_a(mx-n)$, where $m>0$, has a vertical asymptote $x=\frac{n}{m}$ and implied domain $(\frac{n}{m},\mathrm{\infty })$. The x-axis intercept is $1+\frac{n}{m}$.