1. A function $f(x)=a^x,$ with $a>0$ and $a\ne 1,$ is called an exponential function with base a and exponent x.

   Rules of exponents: $a^x{a}^y=a^{x+y},{\displaystyle \frac{a^x}{a^y}}=a^{x-y},$

   $$\begin{array}{l}{(ab)}^x=a^x{b}^x,{\left({\displaystyle \frac{a}{b}}\right)}^x={\displaystyle \frac{a^x}{b^x}}\\ {(a^x)}^y=a^{xy},a^0=1,a^{-x}={\displaystyle \frac{1}{a^x}}\end{array}$$

2. Exponential functions are one-to-one: If $a^u=a^v,$ then $u=v.$

3. If $a>1,$ then $f(x)=a^x$ is an increasing function; $f(x)\to \infty$ as $x\to \infty$ and $f(x)\to 0$ as $x\to -\infty .$

4. If $0<a<1,$ then $f(x)=a^x$ is a decreasing function; $f(x)\to 0$ as $x\to \infty$ and $f(x)\to \infty$ as $x\to -\infty .$

5. The graph of $f(x)=a^x$ has y-intercept 1, and the x-axis is a horizontal asymptote.

6. Simple interest formula. If P dollars is invested at an interest rate r per year for t years, then the simple interest is given by the formula $I=Prt.$ The future value $A(t)=P+Prt.$

7. Compound interest. P dollars invested at an annual rate r compounded n times per year for t years amounts to

   $$A(t)=P{(1+{\displaystyle \frac{r}{n}})}^{nt}.$$

8. The Euler constant $e=\underset{h\to \infty }{\lim }{\left(1+{\displaystyle \frac{1}{h}}\right)}^h\approx \mathrm{2.718.}$

9. Continuous compounding. P dollars invested at an annual rate r compounded continuously for t years amounts to $A=P{e}^{rt}.$

10. The function $f(x)=e^x$ is the natural exponential function.

1. For $x>0,a>0$, and $a\ne 1,y={\log }_{a}x$ if and only if $x=a^y.$

2. Basic properties: ${\log }_{a}a=1,{\log }_{a}1=0,$

   $${\log }_a{a}^x=x,a^{{\log }_{a}x}=x\text{Inverse properties}$$

3. The domain of ${\log }_{a}x$ is $(0,\infty ),$ the range is $(-\infty ,\infty ),$ and the y-axis is a vertical asymptote. The x-intercept is 1.

4. Logarithmic functions are one-to-one: If ${\log }_{a}x={\log }_{a}y,$ then $x=y.$

5. If $a>1,$ then $f(x)={\log }_{a}x$ is an increasing function; $f(x)\to \infty$ as $x\to \infty$ and $f(x)\to -\infty$ as $x\to 0^{+}.$

6. If $0<a<1,$ then $f(x)={\log }_{a}x$ is a decreasing function; $f(x)\to -\infty$ as $x\to \infty$ and $f(x)\to \infty$ as $x\to 0^{+}.$

7. The common logarithmic function is $y=\log x$ (base 10); the natural logarithmic function is $y=\ln x$ (base e).

1. Rules of logarithms:

   $$\begin{array}{rcll}{\log }_a(MN)& =& {\log }_{a}M+{\log }_{a}N& \text{Product rule}\\ {\log }_a\left({\displaystyle \frac{M}{N}}\right)& =& {\log }_{a}M-{\log }_{a}N& \text{Quotient rule}\\ {\log }_a{M}^r& =& r{\log }_{a}M& \text{Power rule}\end{array}$$

2. Change-of-base formula:

   $$\begin{array}{llll}{\log }_{b}x=& {\displaystyle \frac{{\log }_{a}x}{{\log }_{a}b}}& ={\displaystyle \frac{\log x}{\log b}}& ={\displaystyle \frac{\ln x}{\ln b}}\\ & (\text{base }a)& (\text{base }10)& (\text{base }e)\end{array}$$

An exponential equation is an equation in which a variable occurs in one or more exponents.

A logarithmic equation is an equation that involves the logarithm of a function of the variable.

Exponential and logarithmic equations are solved by using some or all of the following techniques:

1. Using the one-to-one property of exponential and logarithmic functions.

2. Converting from exponential to logarithmic form or vice versa

3. Using the product, quotient, and power rules for exponents and logarithms

We use similar techniques to solve logarithmic and exponential inequalities (see page 483).

A logarithmic scale is a scale in which logarithms are used in the measurement of quantities.

1. $p\text{H}=-\log [{\text{H}}^{+}],$ where $[{\text{H}}^{+}]$ is the concentration of ${\text{H}}^{+}$ ions in moles per liter.

2. The Richter scale is used to measure the magnitude M of an earthquake.

   $M=\log \left({\displaystyle \frac{I}{I_0}}\right)$, where I is the intensity of the earthquake and $I_0$ is the zero-level earthquake.

   The energy E released by an earthquake of magnitude M is given by $\log E=4.4+1.5M.$

3. The intensity of a sound wave is defined as the amount of power the wave transmits through a given area. The loudness L of a sound of intensity I is given by $L=10\log \left({\displaystyle \frac{I}{I_0}}\right),$ where $I_0={10}^{-12}\text{W}/{\text{m}}^2$ is the intensity of the threshold of hearing. Equivalently, $I={10}^{L/10}{I}_0.$