Exponential and Logarithmic Functions: Definitions, Properties, Graphs, and Applications

Exponential Functions

Definition of the Exponential Function

An exponential function with base b is defined by the equation $f(x) = b^x$ or $y = b^x$, where b is a positive constant other than 1 ($b > 0$ and $b \neq 1$) and x is any real number.

• Base: The constant $b$.

• Exponent: The variable $x$.

Exponential functions are used to model growth and decay in various real-world contexts, such as population growth, radioactive decay, and compound interest.

Evaluating an Exponential Function

• To evaluate $f(x) = b^x$, substitute the given value of $x$ and compute the result.

• Example: If $f(x) = 42.2(1.56)^x$ models average amount spent after $x$ hours at a shopping mall, then after 3 hours, $f(3) = 42.2(1.56)^3$.

Transformations Involving Exponential Functions

Transformation	Equation	Description
Vertical translation	$f(x) + c$	Shifts the graph up $c$ units.
Vertical translation	$f(x) - c$	Shifts the graph down $c$ units.
Horizontal translation	$f(x + c)$	Shifts the graph left $c$ units.
Horizontal translation	$f(x - c)$	Shifts the graph right $c$ units.
Reflection	$-f(x)$	Reflects the graph about the x-axis.
Reflection	$f(-x)$	Reflects the graph about the y-axis.
Vertical stretch/shrink	$a f(x)$	Vertically stretches if $a > 1$, shrinks if $0 < a < 1$.
Horizontal stretch/shrink	$f(bx)$	Horizontally shrinks if $b > 1$, stretches if $0 < b < 1$.

Graphing Exponential Functions

• To graph $f(x) = b^x$, plot points for several values of $x$ and connect them smoothly.

• The graph passes through $(0,1)$ since $b^0 = 1$.

• The x-axis ($y=0$) is a horizontal asymptote.

• Domain: $(-\infty, \infty)$

• Range: $(0, \infty)$

Characteristics of Exponential Functions of the Form $f(x) = b^x$

• The domain is all real numbers: $(-\infty, \infty)$.

• The range is all positive real numbers: $(0, \infty)$.

• If $b > 1$, the function is increasing; if $0 < b < 1$, the function is decreasing.

• The y-intercept is always $(0,1)$.

• The x-axis is a horizontal asymptote.

The Natural Base $e$

• The number $e$ is an irrational constant approximately equal to 2.71828.

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

Evaluating Functions with Base $e$

• Substitute the value of $x$ into the function and compute $e^x$ using a calculator if necessary.

• Example: $f(x) = 80e^{-0.3x} + 20$ models the percentage of information remembered after $x$ weeks.

Formulas for Compound Interest

• For $n$ compounding periods per year:

$A = P \left(1 + \frac{r}{n}\right)^{nt}$

• For continuous compounding:

$A = Pe^{rt}$

• $A$ = accumulated amount, $P$ = principal, $r$ = annual interest rate (decimal), $n$ = number of compounding periods per year, $t$ = time in years.

Logarithmic Functions

Definition of the Logarithmic Function

For $x > 0$ and $b > 0$, $b \neq 1$, $y = \log_b x$ is equivalent to $b^y = x$. The function $f(x) = \log_b x$ is the logarithmic function with base $b$.

Changing Between Logarithmic and Exponential Forms

• Logarithmic to Exponential: $y = \log_b x \iff b^y = x$

• Exponential to Logarithmic: $b^y = x \iff y = \log_b x$

Evaluating Logarithms

• Find the exponent to which the base must be raised to obtain the given number.

• Example: $\log_{10} 100 = 2$ because $10^2 = 100$.

Properties of Logarithms

• Product Rule: $\log_b(MN) = \log_b M + \log_b N$

• Quotient Rule: $\log_b\left(\frac{M}{N}\right) = \log_b M - \log_b N$

• Power Rule: $\log_b(M^p) = p \log_b M$

• Change-of-Base Property: $\log_b M = \frac{\log_a M}{\log_a b}$

• Inverse Properties: $b^{\log_b x} = x$ and $\log_b(b^x) = x$

Characteristics of Logarithmic Functions of the Form $f(x) = \log_b x$

• Domain: $(0, \infty)$

• Range: $(-\infty, \infty)$

• The y-axis ($x=0$) is a vertical asymptote.

• The graph passes through $(1,0)$ since $\log_b 1 = 0$.

• If $b > 1$, the function is increasing; if $0 < b < 1$, the function is decreasing.

Common and Natural Logarithms

• Common logarithm: Base 10, written as $\log x$.

• Natural logarithm: Base $e$, written as $\ln x$.

Properties of Common and Natural Logarithms

General Properties	Common Logarithms	Natural Logarithms
$\log_b 1 = 0$	$\log 1 = 0$	$\ln 1 = 0$
$\log_b b = 1$	$\log 10 = 1$	$\ln e = 1$
$\log_b b^x = x$	$\log 10^x = x$	$\ln e^x = x$
$b^{\log_b x} = x$	$10^{\log x} = x$	$e^{\ln x} = x$

Expanding and Condensing Logarithmic Expressions

• Use the product, quotient, and power rules to expand or condense logarithmic expressions.

• Example (Expand): $\log_2(7 \cdot 11) = \log_2 7 + \log_2 11$

• Example (Condense): $\log 25 + \log 4 = \log 100$

The Change-of-Base Property

• For any logarithmic bases $a$ and $b$, and any positive number $M$:

$\log_b M = \frac{\log_a M}{\log_a b}$

• Commonly used with $a = 10$ (common log) or $a = e$ (natural log):

$\log_b M = \frac{\log M}{\log b}$ or $\log_b M = \frac{\ln M}{\ln b}$

Exponential and Logarithmic Equations

Solving Exponential Equations by Expressing Each Side as a Power of the Same Base

• If $b^M = b^N$, then $M = N$.

• Rewrite both sides with the same base, set exponents equal, and solve for the variable.

• Example: $5^{2x} = 125 \implies 5^{2x} = 5^3 \implies 2x = 3 \implies x = 1.5$

Using Logarithms to Solve Exponential Equations

1. Isolate the exponential expression.

2. Take the logarithm of both sides (common or natural log).

3. Simplify using properties: $\ln b^x = x \ln b$ or $\log b^x = x \log b$.

4. Solve for the variable.

Solving Logarithmic Equations

• Express the equation in the form $\log_b M = c$.

• Rewrite in exponential form: $M = b^c$.

• Solve for the variable and check for extraneous solutions (argument must be positive).

Using the One-to-One Property of Logarithms

• If $\log_b M = \log_b N$, then $M = N$ (for $M > 0$, $N > 0$).

• Solve for the variable and check the solution in the original equation.

Exponential Growth and Decay

Exponential Growth and Decay Models

• The general model is $A = A_0 e^{kt}$ or $f(t) = A_0 e^{kt}$.

• If $k > 0$, the function models growth; if $k < 0$, it models decay.

• $A_0$ is the initial amount, $k$ is the growth/decay rate, $t$ is time.

Half-Life

• The half-life of a substance is the time required for half of the substance to decay.

• Exponential decay can be modeled as $A = A_0 e^{kt}$, where $k$ is negative.

Logistic Growth Model

• The logistic model for limited growth is $f(t) = \frac{c}{1 + a e^{-bt}}$ or $A = \frac{c}{1 + a e^{-bt}}$, where $c > 0$, $a > 0$, $b > 0$.

• This model describes growth that starts exponentially but levels off as it approaches a maximum value $c$.

Applications

• Exponential and logarithmic functions are used to model population growth, radioactive decay, compound interest, and learning curves.

• For example, the population of Africa or the decay of strontium-90 can be modeled using these functions.

Additional info: The notes include numerous exercises for practice, including graphing, evaluating, and solving equations involving exponential and logarithmic functions, as well as real-world applications such as compound interest and population growth.