Essential Concepts

Exponential Functions

• An exponential function is defined as a function with a positive constant other than 1 raised to a variable exponent. That is: [latex]f\left(x\right)={b}^{x}\text{, where }b>0, b\ne 1[/latex]
• A function is evaluated by solving at a specific input value.
• An exponential model can be found when the growth rate and initial value are known.
• An exponential model can be found when two data points from the model are known.
• An exponential model can be found using two data points from the graph of the model.
• The graph of the function [latex]f\left(x\right)={b}^{x}[/latex] has a [latex]y[/latex]–intercept at [latex]\left(0, 1\right)[/latex], domain of [latex]\left(-\infty , \infty \right)[/latex], range of [latex]\left(0, \infty \right)[/latex], and horizontal asymptote of [latex]y=0[/latex].
• If [latex]b>1[/latex], the function is increasing. The left tail of the graph will approach the asymptote [latex]y=0[/latex], and the right tail will increase without bound. This is called an exponential growth.
• If [latex]0 < b < 1[/latex], the function is decreasing. The left tail of the graph will increase without bound, and the right tail will approach the asymptote [latex]y=0[/latex]. This is called an exponential decay.
• The equation [latex]f\left(x\right)={b}^{x} d[/latex] represents a vertical shift of the parent function [latex]f\left(x\right)={b}^{x}[/latex].
• The equation [latex]f\left(x\right)={b}^{x c}[/latex] represents a horizontal shift of the parent function [latex]f\left(x\right)={b}^{x}[/latex].
• The equation [latex]f\left(x\right)=a{b}^{x}[/latex], where [latex]a>0[/latex], represents a vertical stretch if [latex]|a|>1[/latex] or compression if [latex]0<|a|<1[/latex] of the parent function [latex]f\left(x\right)={b}^{x}[/latex].
• When the parent function [latex]f\left(x\right)={b}^{x}[/latex] is multiplied by [latex]–1[/latex], the result, [latex]f\left(x\right)=-{b}^{x}[/latex], is a reflection about the x-axis. When the input is multiplied by –1, the result, [latex]f\left(x\right)={b}^{-x}[/latex], is a reflection about the [latex]y[/latex]-axis.
• All transformations of the exponential function can be summarized by the general equation [latex]f\left(x\right)=a{b}^{x c} d[/latex].
• The value of an account at any time [latex]t[/latex] can be calculated using the compound interest formula when the principal, annual interest rate, and compounding periods are known.
• The initial investment of an account can be found using the compound interest formula when the value of the account, annual interest rate, compounding periods, and life span of the account are known.
• The number [latex]e[/latex] is a mathematical constant often used as the base of real world exponential growth and decay models. Its decimal approximation is [latex]e\approx 2.718282[/latex].
• Continuous growth or decay models are exponential models that use [latex]e[/latex] as the base. Continuous growth and decay models can be found when the initial value and growth or decay rate are known.

Logarithmic Functions

• The inverse of an exponential function is a logarithmic function, and the inverse of a logarithmic function is an exponential function.
• Logarithmic equations can be written in an equivalent exponential form using the definition of a logarithm.
• Exponential equations can be written in an equivalent logarithmic form using the definition of a logarithm.
• For [latex]\text{ } x>0,b>0,b\ne 1[/latex], [latex]y={\mathrm{log}}_{b}\left(x\right)[/latex] if and only if [latex]\text{ }{b}^{y}=x[/latex].
• Logarithmic functions with base [latex]b[/latex] can be evaluated mentally using previous knowledge of powers of [latex]b[/latex].
• Common logarithms is a logarithmic function with base [latex]10[/latex].
• Natural logarithm is a logarithmic function with base [latex]3[/latex].
• To find the domain of a logarithmic function, set up an inequality showing the argument greater than zero and solve for [latex]x[/latex].
• The graph of the parent function [latex]f\left(x\right)={\mathrm{log}}_{b}\left(x\right)[/latex] has an [latex]x[/latex]–intercept at [latex]\left(1,0\right)[/latex], domain [latex]\left(0,\infty \right)[/latex], range [latex]\left(-\infty ,\infty \right)[/latex], vertical asymptote [latex]x = 0[/latex], and
  • if [latex]b > 1[/latex], the function is increasing.
  • if [latex]0 < b < 1[/latex], the function is decreasing.
• The equation [latex]f\left(x\right)={\mathrm{log}}_{b}\left(x c\right)[/latex] shifts the parent function [latex]y={\mathrm{log}}_{b}\left(x\right)[/latex] horizontally
  • left [latex]c[/latex] units if [latex]c > 0[/latex].
  • right [latex]c[/latex] units if [latex]c < 0[/latex].
• The equation [latex]f\left(x\right)={\mathrm{log}}_{b}\left(x\right) d[/latex] shifts the parent function [latex]y={\mathrm{log}}_{b}\left(x\right)[/latex] vertically
  • up [latex]d[/latex] units if [latex]d > 0[/latex].
  • down [latex]d[/latex] units if [latex]d < 0[/latex].
• For any constant [latex]a > 0[/latex], the equation [latex]f\left(x\right)=a{\mathrm{log}}_{b}\left(x\right)[/latex]
  • stretches the parent function [latex]y={\mathrm{log}}_{b}\left(x\right)[/latex] vertically by a factor of [latex]a[/latex] if [latex]|a| > 1[/latex].
  • compresses the parent function [latex]y={\mathrm{log}}_{b}\left(x\right)[/latex] vertically by a factor of [latex]a[/latex] if [latex]|a| < 1[/latex].
• When the parent function [latex]y={\mathrm{log}}_{b}\left(x\right)[/latex] is multiplied by [latex]–1[/latex], the result is a reflection about the x-axis. When the input is multiplied by[latex]–1[/latex], the result is a reflection about the [latex]y[/latex]-axis.
  • The equation [latex]f\left(x\right)=-{\mathrm{log}}_{b}\left(x\right)[/latex] represents a reflection of the parent function about the [latex]x[/latex]–axis.
  • The equation [latex]f\left(x\right)={\mathrm{log}}_{b}\left(-x\right)[/latex] represents a reflection of the parent function about the [latex]y[/latex]–axis.
  • A graphing calculator may be used to approximate solutions to some logarithmic equations.
• All transformations of the logarithmic function can be summarized by the general equation [latex]f\left(x\right)=a{\mathrm{log}}_{b}\left(x c\right) d[/latex].
• To find the domain of a logarithmic function, set up an inequality showing the argument greater than zero and solve for [latex]x[/latex].

Key Equations

definition of the exponential function	[latex]f\left(x\right)={b}^{x}\text{, where }b>0, b\ne 1[/latex]
definition of exponential growth	[latex]f\left(x\right)=a{b}^{x},\text{ where }a>0,b>0,b\ne 1[/latex]
compound interest formula	[latex]\begin{array}{l}A\left(t\right)=P{\left(1 \frac{r}{n}\right)}^{nt} ,\text{ where}\hfill \\ A\left(t\right)\text{ is the account value at time }t\hfill \\ t\text{ is the number of years}\hfill \\ P\text{ is the initial investment, often called the principal}\hfill \\ r\text{ is the annual percentage rate (APR), or nominal rate}\hfill \\ n\text{ is the number of compounding periods in one year}\hfill \end{array}[/latex]
continuous growth formula	[latex]A\left(t\right)=a{e}^{rt},\text{ where }[/latex]t is the number of time periods of growth [latex]\\[/latex]a is the starting amount (in the continuous compounding formula a is replaced with P, the principal)[latex]\\[/latex]e is the mathematical constant, [latex]e\approx 2.718282[/latex]
General Form for the Transformation of the Parent Function [latex]\text{ }f\left(x\right)={b}^{x}[/latex]	[latex]f\left(x\right)=a{b}^{x c} d[/latex]
General Form for the Transformation of the Parent Logarithmic Function [latex]\text{ }f\left(x\right)={\mathrm{log}}_{b}\left(x\right)[/latex]	[latex]f\left(x\right)=a{\mathrm{log}}_{b}\left(x c\right) d[/latex]

Glossary

annual percentage rate (APR)

the yearly interest rate earned by an investment account, also called nominal rate

common logarithm

the exponent to which [latex]10[/latex] must be raised to get [latex]x[/latex]; [latex]{\mathrm{log}}_{10}\left(x\right)[/latex] is written simply as [latex]\mathrm{log}\left(x\right)[/latex]

compound interest

interest earned on the total balance, not just the principal

exponential growth

a model that grows by a rate proportional to the amount present

logarithm

the exponent to which [latex]b[/latex] must be raised to get [latex]x[/latex]; written [latex]y={\mathrm{log}}_{b}\left(x\right)[/latex]

natural logarithm

the exponent to which the number [latex]e[/latex] must be raised to get [latex]x[/latex]; [latex]{\mathrm{log}}_{e}\left(x\right)[/latex] is written as [latex]\mathrm{ln}\left(x\right)[/latex]

nominal rate

the yearly interest rate earned by an investment account, also called annual percentage rate