### Key Concepts

## 10.1 Finding Composite and Inverse Functions

**Composition of Functions:**The composition of functions $f$ and $g,$ is written $f\circ g$ and is defined by

$$\left(f\circ g\right)\left(x\right)=f\left(g\left(x\right)\right)$$

We read $f\left(g\left(x\right)\right)$ as $f$ of $g$ of $x.$**Horizontal Line Test:**If every horizontal line, intersects the graph of a function in at most one point, it is a one-to-one function.**Inverse of a Function Defined by Ordered Pairs:**If $f\left(x\right)$ is a one-to-one function whose ordered pairs are of the form $\left(x,y\right),$ then its inverse function ${f}^{\mathrm{-1}}\left(x\right)$ is the set of ordered pairs $\left(y,x\right).$**Inverse Functions:**For every $x$ in the domain of one-to-one function $f$ and ${f}^{\mathrm{-1}},$

$$\begin{array}{ccc}\hfill {f}^{\mathrm{-1}}\left(f\left(x\right)\right)& =\hfill & x\hfill \\ \hfill f\left({f}^{\mathrm{-1}}\left(x\right)\right)& =\hfill & x\hfill \end{array}$$**How to Find the Inverse of a One-to-One Function:**- Step 1.
Substitute
*y*for $f\left(x\right).$ - Step 2.
Interchange the variables
*x*and*y*. - Step 3.
Solve for
*y*. - Step 4. Substitute ${f}^{\mathrm{-1}}\left(x\right)$ for $y.$
- Step 5. Verify that the functions are inverses.

- Step 1.
Substitute

## 10.2 Evaluate and Graph Exponential Functions

**Properties of the Graph of**$f\left(x\right)={a}^{x}:$

when $a>1$ when $0<a<1$ Domain $\left(\text{\u2212}\infty ,\infty \right)$ Domain $\left(\text{\u2212}\infty ,\infty \right)$ Range $\left(0,\infty \right)$ Range $\left(0,\infty \right)$ $x$-intercept none $x$-intercept none $y$-intercept $\left(0,1\right)$ $y$-intercept $\left(0,1\right)$ Contains $\left(1,a\right),\phantom{\rule{0.2em}{0ex}}\left(\mathrm{-1},\frac{1}{a}\right)$ Contains $\left(1,a\right),\phantom{\rule{0.2em}{0ex}}\left(\mathrm{-1},\frac{1}{a}\right)$ Asymptote $x$-axis, the line $y=0$ Asymptote $x$-axis, the line $y=0$ Basic shape increasing Basic shape decreasing

**One-to-One Property of Exponential Functions:**

For $a>0$ and $a\ne 1,$

$$\text{If}\phantom{\rule{0.2em}{0ex}}{a}^{x}={a}^{y},\phantom{\rule{0.2em}{0ex}}\text{then}\phantom{\rule{0.2em}{0ex}}x=y.$$**How to Solve an Exponential Equation**- Step 1. Write both sides of the equation with the same base, if possible.
- Step 2. Write a new equation by setting the exponents equal.
- Step 3. Solve the equation.
- Step 4. Check the solution.

**Compound Interest:**For a principal, $P,$ invested at an interest rate, $r,$ for $t$ years, the new balance, $A,$ is

$\begin{array}{}\\ \\ A=P{\left(1+\frac{r}{n}\right)}^{nt}\hfill & & & & & \text{when compounded}\phantom{\rule{0.2em}{0ex}}n\phantom{\rule{0.2em}{0ex}}\text{times a year.}\hfill \\ A=P{e}^{rt}\hfill & & & & & \text{when compounded continuously.}\hfill \end{array}$**Exponential Growth and Decay:**For an original amount, ${A}_{0}$ that grows or decays at a rate, $r,$ for a certain time $t,$ the final amount,$A,$ is $A={A}_{0}{e}^{rt}.$

## 10.3 Evaluate and Graph Logarithmic Functions

**Properties of the Graph of**$y={\text{log}}_{a}x:$

**when**$a>1$**when**$0<a<1$Domain $\left(0,\infty \right)$ Domain $\left(0,\infty \right)$ Range $\left(\text{\u2212}\infty ,\infty \right)$ Range $\left(\text{\u2212}\infty ,\infty \right)$ *x*-intercept$\left(1,0\right)$ *x*-intercept$\left(1,0\right)$ *y*-interceptnone *y*-interceptnone Contains $\left(a,1\right),$$\left(\frac{1}{a},\mathrm{-1}\right)$ Contains $\left(a,1\right),$$\left(\frac{1}{a},\mathrm{-1}\right)$ Asymptote *y*-axisAsymptote *y*-axisBasic shape increasing Basic shape decreasing

**Decibel Level of Sound:**The loudness level, $D$, measured in decibels, of a sound of intensity, $I$, measured in watts per square inch is $D=10\text{log}\left(\frac{I}{{10}^{\mathrm{-12}}}\right).$**Earthquake Intensity:**The magnitude $R$ of an earthquake is measured by $R=\text{log}\phantom{\rule{0.2em}{0ex}}I,$ where $I$ is the intensity of its shock wave.

## 10.4 Use the Properties of Logarithms

**Properties of Logarithms**

$$\phantom{\rule{3em}{0ex}}{\text{log}}_{a}1=0\phantom{\rule{5em}{0ex}}{\text{log}}_{a}a=1$$**Inverse Properties of Logarithms**- For $a>0,$$x>0$ and $a\ne 1$

$${a}^{{\text{log}}_{a}x}=x\phantom{\rule{4.95em}{0ex}}{\text{log}}_{a}{a}^{x}=x$$

- For $a>0,$$x>0$ and $a\ne 1$
**Product Property of Logarithms**

- If $M>0,N>0\text{,}\phantom{\rule{0.2em}{0ex}}\text{a}>0$ and $\text{a}\ne 1,$ then,

$$\phantom{\rule{2.08em}{0ex}}{\text{log}}_{a}M\xb7N={\text{log}}_{a}M+{\text{log}}_{a}N$$

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

- If $M>0,N>0\text{,}\phantom{\rule{0.2em}{0ex}}\text{a}>0$ and $\text{a}\ne 1,$ then,
**Quotient Property of Logarithms**

- If $M>0,N>0\text{,}\phantom{\rule{0.2em}{0ex}}\text{a}>0$ and $\text{a}\ne 1,$ then,

$$\phantom{\rule{3em}{0ex}}{\text{log}}_{a}\frac{M}{N}={\text{log}}_{a}M-{\text{log}}_{a}N$$

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

- If $M>0,N>0\text{,}\phantom{\rule{0.2em}{0ex}}\text{a}>0$ and $\text{a}\ne 1,$ then,
**Power Property of Logarithms**

- If $M>0,\phantom{\rule{0.2em}{0ex}}\text{a}>0,\phantom{\rule{0.2em}{0ex}}\text{a}\ne 1$ and $p$ is any real number then,

$${\text{log}}_{a}{M}^{p}=p\phantom{\rule{0.2em}{0ex}}{\text{log}}_{a}M$$

The log of a number raised to a power is the product of the power times the log of the number.

- If $M>0,\phantom{\rule{0.2em}{0ex}}\text{a}>0,\phantom{\rule{0.2em}{0ex}}\text{a}\ne 1$ and $p$ is any real number then,
**Properties of Logarithms Summary**

If $M>0,\phantom{\rule{0.2em}{0ex}}\text{a}>0,\phantom{\rule{0.2em}{0ex}}\text{a}\ne 1$ and $p$ is any real number then,

Property Base $a$ Base $e$ ${\text{log}}_{a}1=0$ $\text{ln}1=0$ ${\text{log}}_{a}a=1$ $\text{ln}\phantom{\rule{0.2em}{0ex}}e=1$ **Inverse Properties**$\begin{array}{}\\ \\ \phantom{\rule{1.25em}{0ex}}{a}^{{\text{log}}_{a}x}=x\hfill \\ \phantom{\rule{1em}{0ex}}{\text{log}}_{a}{a}^{x}=x\hfill \end{array}$ $\begin{array}{}\\ \\ \phantom{\rule{1.2em}{0ex}}{e}^{\text{ln}\phantom{\rule{0.2em}{0ex}}x}=x\hfill \\ \phantom{\rule{1em}{0ex}}\text{ln}\phantom{\rule{0.2em}{0ex}}{e}^{x}=x\hfill \end{array}$ **Product Property of Logarithms**${\text{log}}_{a}\left(M\xb7N\right)={\text{log}}_{a}M+{\text{log}}_{a}N$ $\text{ln}\left(M\phantom{\rule{0.2em}{0ex}}\xb7\phantom{\rule{0.2em}{0ex}}N\right)=\text{ln}\phantom{\rule{0.2em}{0ex}}M+\text{ln}\phantom{\rule{0.2em}{0ex}}N$ **Quotient Property of Logarithms**$\phantom{\rule{1.9em}{0ex}}{\text{log}}_{a}\frac{M}{N}={\text{log}}_{a}M-{\text{log}}_{a}N$ $\phantom{\rule{2.4em}{0ex}}\text{ln}\frac{M}{N}=\text{ln}\phantom{\rule{0.2em}{0ex}}M-\text{ln}\phantom{\rule{0.2em}{0ex}}N$ **Power Property of Logarithms**$\phantom{\rule{1.55em}{0ex}}{\text{log}}_{a}{M}^{p}=p\phantom{\rule{0.2em}{0ex}}{\text{log}}_{a}M$ $\phantom{\rule{1.76em}{0ex}}\text{ln}\phantom{\rule{0.2em}{0ex}}{M}^{p}=p\text{ln}\phantom{\rule{0.2em}{0ex}}M$ **Change-of-Base Formula**

For any logarithmic bases*a*and*b*, and $M>0,$

$$\begin{array}{ccccccc}\hfill {\text{log}}_{a}M=\frac{{\text{log}}_{b}M}{{\text{log}}_{b}a}\hfill & & & \hfill \phantom{\rule{2em}{0ex}}{\text{log}}_{a}M=\frac{\text{log}M}{\text{log}a}\hfill & & & \hfill \phantom{\rule{2em}{0ex}}{\text{log}}_{a}M=\frac{\text{ln}\phantom{\rule{0.2em}{0ex}}M}{\text{ln}\phantom{\rule{0.2em}{0ex}}a}\hfill \\ \hfill \text{new base}\phantom{\rule{0.2em}{0ex}}b\hfill & & & \hfill \phantom{\rule{2em}{0ex}}\text{new base 10}\hfill & & & \hfill \phantom{\rule{2em}{0ex}}\text{new base}\phantom{\rule{0.2em}{0ex}}e\hfill \end{array}$$

## 10.5 Solve Exponential and Logarithmic Equations

**One-to-One Property of Logarithmic Equations:**For $M>0,N>0,\phantom{\rule{0.2em}{0ex}}a\phantom{\rule{0.2em}{0ex}}\text{>}\phantom{\rule{0.2em}{0ex}}0,$ and $\text{a}\ne 1$ is any real number:

$$\text{If}\phantom{\rule{0.2em}{0ex}}{\text{log}}_{a}M={\text{log}}_{a}N,\phantom{\rule{0.2em}{0ex}}\text{then}\phantom{\rule{0.2em}{0ex}}M=N.$$**Compound Interest:**

For a principal,*P*, invested at an interest rate,*r*, for*t*years, the new balance,*A*, is:

$$\begin{array}{}\\ \\ \phantom{\rule{5em}{0ex}}A=P{\left(1+\frac{r}{n}\right)}^{nt}\hfill & & & & & \text{when compounded}\phantom{\rule{0.2em}{0ex}}n\phantom{\rule{0.2em}{0ex}}\text{times a year.}\hfill \\ \phantom{\rule{5em}{0ex}}A=P{e}^{rt}\hfill & & & & & \text{when compounded continuously.}\hfill \end{array}$$**Exponential Growth and Decay:**For an original amount, ${A}_{0}$ that grows or decays at a rate,*r*, for a certain time*t*, the final amount,*A*, is $A={A}_{0}{e}^{rt}.$