Essential Concepts
Graphs of Exponential Functions
Characteristics of the Parent Function [latex]f(x) = b^x[/latex]:
For [latex]b > 0[/latex], [latex]b \ne 1[/latex]:
- Domain: [latex](-\infty, \infty)[/latex]
- Range: [latex](0, \infty)[/latex]
- Horizontal asymptote: [latex]y = 0[/latex]
- y-intercept: [latex](0, 1)[/latex]
Exponential Growth ([latex]b > 1[/latex]):
- As [latex]x \rightarrow \infty[/latex], [latex]f(x) \rightarrow \infty[/latex]
- As [latex]x \rightarrow -\infty[/latex], [latex]f(x) \rightarrow 0[/latex]
Exponential Decay ([latex]0 < b < 1[/latex]):
- As [latex]x \rightarrow \infty[/latex], [latex]f(x) \rightarrow 0[/latex]
- As [latex]x \rightarrow -\infty[/latex], [latex]f(x) \rightarrow \infty[/latex]
Transformations of Exponential Graphs:
General form: [latex]f(x) = ab^{x+c} + d[/latex]
| Transformation | Form | Effect on Graph |
|---|---|---|
| Vertical shift | [latex]f(x) = b^x + d[/latex] | Shifts up [latex]d[/latex] units if [latex]d > 0[/latex], down if [latex]d < 0[/latex]; asymptote becomes [latex]y = d[/latex]; range becomes [latex](d, \infty)[/latex] |
| Horizontal shift | [latex]f(x) = b^{x+c}[/latex] | Shifts left [latex]c[/latex] units if [latex]c > 0[/latex], right if [latex]c < 0[/latex]; y-intercept becomes [latex](0, b^c)[/latex] |
| Vertical stretch | [latex]f(x) = ab^x[/latex] where [latex]|a| > 1[/latex] | Stretches vertically by factor of [latex]|a|[/latex]; y-intercept becomes [latex](0, a)[/latex] |
| Vertical compression | [latex]f(x) = ab^x[/latex] where [latex]0 < |a| < 1[/latex] | Compresses vertically by factor of [latex]|a|[/latex]; y-intercept becomes [latex](0, a)[/latex] |
| Reflection over x-axis | [latex]f(x) = -b^x[/latex] | Reflects across x-axis; range becomes [latex](-\infty, 0)[/latex]; y-intercept becomes [latex](0, -1)[/latex] |
| Reflection over y-axis | [latex]f(x) = b^{-x}[/latex] | Reflects across y-axis; equivalent to [latex]f(x) = \left(\frac{1}{b}\right)^x[/latex]; reverses growth/decay |
Exponential Functions
General Form: [latex]f(x) = ab^x[/latex] where:
- [latex]a[/latex] is the initial value (the output when [latex]x = 0[/latex])
- [latex]b[/latex] is the base (growth factor or decay factor)
- [latex]b > 0[/latex] and [latex]b \ne 1[/latex]
The Number [latex]e[/latex]:
The mathematical constant [latex]e \approx 2.718282[/latex] is defined as: [latex]e = \lim_{n \to \infty}\left(1 + \frac{1}{n}\right)^n[/latex]
The natural exponential function [latex]f(x) = e^x[/latex] has the same characteristics as other exponential functions with base [latex]b > 1[/latex].
Finding Exponential Equations from Two Points:
Case 1: If one point is [latex](0, a)[/latex]:
- [latex]a[/latex] is the initial value
- Substitute the second point into [latex]f(x) = ab^x[/latex]
- Solve for [latex]b[/latex]
Case 2: If neither point is [latex](0, a)[/latex]:
- Substitute both points into [latex]f(x) = ab^x[/latex]
- Create a system of two equations
- Solve for [latex]a[/latex] and [latex]b[/latex]
Finding Equations from Graphs:
- Identify two points on the graph (preferably including the y-intercept)
- Use the y-intercept as the initial value [latex]a[/latex] if possible
- Substitute another point and solve for [latex]b[/latex]
- Write the equation [latex]f(x) = ab^x[/latex]
Logarithmic Functions
For [latex]x > 0[/latex], [latex]b > 0[/latex], [latex]b \ne 1[/latex]:
[latex]y = \log_b(x)[/latex] means [latex]b^y = x[/latex]
The logarithm [latex]y[/latex] is the exponent to which [latex]b[/latex] must be raised to get [latex]x[/latex].
Converting Between Forms:
| Logarithmic Form | Exponential Form |
|---|---|
| [latex]\log_b(x) = y[/latex] | [latex]b^y = x[/latex] |
| [latex]\log_2(8) = 3[/latex] | [latex]2^3 = 8[/latex] |
| [latex]\log_5(25) = 2[/latex] | [latex]5^2 = 25[/latex] |
| [latex]\log_{10}\left(\frac{1}{10000}\right) = -4[/latex] | [latex]10^{-4} = \frac{1}{10000}[/latex] |
Cannot take the logarithm of a negative number or zero
Common Logarithm: Base 10
- Written as [latex]\log(x)[/latex] instead of [latex]\log_{10}(x)[/latex]
- [latex]y = \log(x)[/latex] is equivalent to [latex]10^y = x[/latex]
Natural Logarithm: Base [latex]e[/latex]
- Written as [latex]\ln(x)[/latex] instead of [latex]\log_e(x)[/latex]
- [latex]y = \ln(x)[/latex] is equivalent to [latex]e^y = x[/latex]
Evaluating Logarithms:
To evaluate [latex]\log_b(x)[/latex], ask: “To what exponent must [latex]b[/latex] be raised to get [latex]x[/latex]?”
Examples:
- [latex]\log_2(8) = 3[/latex] because [latex]2^3 = 8[/latex]
- [latex]\log_7(49) = 2[/latex] because [latex]7^2 = 49[/latex]
- [latex]\log_3\left(\frac{1}{27}\right) = -3[/latex] because [latex]3^{-3} = \frac{1}{27}[/latex]
Since logarithmic and exponential functions are inverses:
- [latex]\log_b(b^x) = x[/latex] for all [latex]x[/latex]
- [latex]b^{\log_b(x)} = x[/latex] for [latex]x > 0[/latex]
- [latex]\ln(e^x) = x[/latex] and [latex]e^{\ln(x)} = x[/latex] for [latex]x > 0[/latex]
- [latex]\log(10^x) = x[/latex] and [latex]10^{\log(x)} = x[/latex] for [latex]x > 0[/latex]
Graphs of Logarithmic Functions
Characteristics of the Parent Function [latex]f(x) = \log_b(x)[/latex]:
For [latex]b > 0[/latex], [latex]b \ne 1[/latex]:
- Domain: [latex](0, \infty)[/latex]
- Range: [latex](-\infty, \infty)[/latex]
- Vertical asymptote: [latex]x = 0[/latex]
- x-intercept: [latex](1, 0)[/latex]
Finding the Domain:
The argument of a logarithm must be positive. For [latex]f(x) = \log_b(\text{expression})[/latex]:
- Set the expression [latex]> 0[/latex]
- Solve the inequality for [latex]x[/latex]
- Write the domain in interval notation
Transformations of Logarithmic Graphs:
General form: [latex]f(x) = a\log_b(x + c) + d[/latex]
| Transformation | Form | Effect on Graph |
|---|---|---|
| Horizontal shift | [latex]f(x) = \log_b(x + c)[/latex] | Shifts left [latex]c[/latex] units if [latex]c > 0[/latex], right if [latex]c < 0[/latex]; asymptote becomes [latex]x = -c[/latex]; domain becomes [latex](-c, \infty)[/latex] |
| Vertical shift | [latex]f(x) = \log_b(x) + d[/latex] | Shifts up [latex]d[/latex] units if [latex]d > 0[/latex], down if [latex]d < 0[/latex]; asymptote remains [latex]x = 0[/latex] |
| Vertical stretch | [latex]f(x) = a\log_b(x)[/latex] where [latex]|a| > 1[/latex] | Stretches vertically by factor of [latex]|a|[/latex]; x-intercept remains [latex](1, 0)[/latex] |
| Vertical compression | [latex]f(x) = a\log_b(x)[/latex] where [latex]0 < |a| < 1[/latex] | Compresses vertically by factor of [latex]|a|[/latex]; x-intercept remains [latex](1, 0)[/latex] |
| Reflection over x-axis | [latex]f(x) = -\log_b(x)[/latex] | Reflects across x-axis; domain [latex](0, \infty)[/latex] and asymptote [latex]x = 0[/latex] unchanged |
| Reflection over y-axis | [latex]f(x) = \log_b(-x)[/latex] | Reflects across y-axis; domain becomes [latex](-\infty, 0)[/latex]; asymptote remains [latex]x = 0[/latex] |
Key Equations
| Description | Equation |
|---|---|
| General exponential function | [latex]f(x) = ab^x[/latex] |
| Transformed exponential function | [latex]f(x) = ab^{x+c} + d[/latex] |
| Natural exponential function | [latex]f(x) = e^x[/latex] where [latex]e \approx 2.718282[/latex] |
| General logarithmic function | [latex]f(x) = \log_b(x)[/latex] |
| Transformed logarithmic function | [latex]f(x) = a\log_b(x + c) + d[/latex] |
| Common logarithm | [latex]\log(x) = \log_{10}(x)[/latex] |
| Natural logarithm | [latex]\ln(x) = \log_e(x)[/latex] |
| Logarithmic-exponential equivalence | [latex]y = \log_b(x) \Leftrightarrow b^y = x[/latex] |
| Inverse property (exponential to log) | [latex]\log_b(b^x) = x[/latex] |
| Inverse property (log to exponential) | [latex]b^{\log_b(x)} = x[/latex] for [latex]x > 0[/latex] |
Glossary
base
The constant [latex]b[/latex] in an exponential function [latex]f(x) = ab^x[/latex] or in a logarithmic function [latex]f(x) = \log_b(x)[/latex]; must be positive and not equal to 1.
common logarithm
A logarithm with base 10, written as [latex]\log(x)[/latex] instead of [latex]\log_{10}(x)[/latex]; used to measure phenomena like earthquakes (Richter scale), star brightness, and pH levels.
exponential decay
A decrease based on a constant multiplicative rate of change over equal increments of time; occurs when [latex]0 < b < 1[/latex] in [latex]f(x) = ab^x[/latex].
exponential function
A function of the form [latex]f(x) = ab^x[/latex] where [latex]a[/latex] is any nonzero number, [latex]b > 0[/latex], and [latex]b \ne 1[/latex].
exponential growth
An increase based on a constant multiplicative rate of change over equal increments of time; occurs when [latex]b > 1[/latex] in [latex]f(x) = ab^x[/latex].
growth factor
The value [latex]b[/latex] in an exponential growth function [latex]f(x) = ab^x[/latex] where [latex]b > 1[/latex]; also called the base or growth multiplier.
horizontal asymptote
A horizontal line that the graph of a function approaches as the input approaches [latex]\infty[/latex] or [latex]-\infty[/latex]; for exponential functions [latex]f(x) = ab^{x+c} + d[/latex], the horizontal asymptote is [latex]y = d[/latex].
initial value
The value [latex]a[/latex] in an exponential function [latex]f(x) = ab^x[/latex]; represents the output when [latex]x = 0[/latex], so [latex]f(0) = a[/latex].
logarithm
The exponent to which a base must be raised to produce a given number; if [latex]b^y = x[/latex], then [latex]\log_b(x) = y[/latex].
logarithmic function
The inverse of an exponential function; a function of the form [latex]f(x) = \log_b(x)[/latex] where [latex]b > 0[/latex] and [latex]b \ne 1[/latex].
natural exponential function
The exponential function with base [latex]e[/latex]; written as [latex]f(x) = e^x[/latex] where [latex]e \approx 2.718282[/latex].
natural logarithm
A logarithm with base [latex]e[/latex], written as [latex]\ln(x)[/latex] instead of [latex]\log_e(x)[/latex]; commonly used in calculus and scientific applications.
one-to-one function
A function in which each output value corresponds to exactly one input value; both exponential and logarithmic functions are one-to-one.
vertical asymptote
A vertical line that the graph of a function approaches but never touches or crosses; for logarithmic functions [latex]f(x) = \log_b(x + c)[/latex], the vertical asymptote is [latex]x = -c[/latex].