Exponential and Logarithmic Functions: Precalculus Study Guide

Expoznctions

Definition and Properties

Exponential functions are fundamental in precalculus, modeling growth and decay processes. A function of the form f(x) = ax (where a > 0 and a ≠ 1) is called an exponential function with base a.

• Domain: All real numbers, $(-\infty, \infty)$

• Range: All real numbers greater than 0, $(0, \infty)$

• Key Properties:

  • $a^{m} \cdot a^{n} = a^{m+n}$

  • $\frac{a^{m}}{a^{n}} = a^{m-n}$

  • $(a^{m})^{n} = a^{mn}$

  • $a^{-n} = \frac{1}{a^{n}}$

  • $a^{0} = 1$

Example: Evaluate $f(x) = 3^{x}$ for $x = 4$:

• $f(4) = 3^{4} = 81$

Graphing Exponential Functions

Exponential functions can be graphed by evaluating the function at various points and plotting the results. The graph of $f(x) = a^{x}$ is always increasing for $a > 1$ and decreasing for $0 < a < 1$.

Transformations of Exponential Functions

Transformations such as shifts and reflections can be applied to exponential functions. For example, $f(x) = a^{-x}$ is the graph of $f(x) = a^{x}$ reflected about the y-axis.

Logarithmic Functions

Definition and Properties

A logarithmic function is the inverse of an exponential function. The function $f(x) = \log_{a}(x)$ (where $a > 0$ and $a \neq 1$) is called the logarithmic function with base a.

• Domain: $(0, \infty)$

• Range: $(-\infty, \infty)$

• Inverse Property: $a^{x} = y \iff x = \log_{a}(y)$

Graphs of Logarithmic Functions

The graph of $f(x) = \log_{a}(x)$ is obtained by reflecting the graph of $f(x) = a^{x}$ about the line $y = x$. The graph is increasing for $a > 1$ and decreasing for $0 < a < 1$.

Properties of Logarithmic Functions

• Domain: $(0, \infty)$

• Range: $(-\infty, \infty)$

• Vertical Asymptote: $x = 0$

• Transformations: Shifts and reflections can be applied to logarithmic functions.

Common and Natural Logarithms

If the base of a logarithm is 10, it is called a common logarithm ($\log_{10}(x)$). If the base is $e$ (Euler's number, approximately 2.71828), it is called a natural logarithm ($\ln(x)$).

• Common Logarithm: $\log(x) = \log_{10}(x)$

• Natural Logarithm: $\ln(x) = \log_{e}(x)$

Rules of Logarithms

Product, Quotient, and Power Rules

Logarithms follow several important rules:

• Product Rule: $\log_{a}(MN) = \log_{a}(M) + \log_{a}(N)$

• Quotient Rule: $\log_{a}(\frac{M}{N}) = \log_{a}(M) - \log_{a}(N)$

• Power Rule: $\log_{a}(M^{r}) = r \log_{a}(M)$

Expanding and Condensing Logarithmic Expressions

Logarithmic expressions can be expanded or condensed using the above rules.

• Example (Expand): $\log_{a}(\frac{x^{2}(2x+1)^{4}}{y}) = \log_{a}(x^{2}) + \log_{a}((2x+1)^{4}) - \log_{a}(y)$

• Example (Condense): $\log_{a}(x) + \log_{a}(y) = \log_{a}(xy)$

Change of Base Formula

The change of base formula allows us to convert logarithms to a different base:

• $\log_{a}(x) = \frac{\log_{b}(x)}{\log_{b}(a)}$

This is commonly used to convert logarithms to base 10 or base $e$ for calculator computations.

Solving Exponential and Logarithmic Equations

Solving Exponential Equations

To solve exponential equations, use properties of exponents or logarithms:

• If possible, rewrite both sides with the same base and use the one-to-one property.

• If not possible, take the logarithm of both sides and solve for the variable.

Example: Solve $5^{x} = 17$:

• Take natural log: $\ln(5^{x}) = \ln(17)$

• Apply power rule: $x \ln(5) = \ln(17)$

• Solve: $x = \frac{\ln(17)}{\ln(5)}$

Solving Logarithmic Equations

To solve logarithmic equations, isolate the logarithm and exponentiate both sides:

• Example: $\log_{2}(x-1) = 3$

• Exponentiate: $x-1 = 2^{3}$

• Solve: $x = 9$

Applications: Exponential Growth, Decay, and Compound Interest

Exponential Growth and Decay

Exponential functions model population growth, radioactive decay, and other processes. The general formula is:

• $A(t) = A_{0} e^{kt}$

• $A_{0}$: Initial amount

• $k$: Growth ($k > 0$) or decay ($k < 0$) rate

Half-Life

The half-life of a substance is the time required for its amount to decrease to half its initial value. The formula is:

• $h = \frac{\ln(2)}{|k|}$

Compound Interest

Compound interest is calculated using exponential functions. The formula for compound interest is:

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

• $A$: Amount after $t$ years

• $P$: Principal amount

• $r$: Annual interest rate (decimal)

• $n$: Number of times interest is compounded per year

• $t$: Number of years

For continuous compounding:

• $A = P e^{rt}$

The Natural Exponential Function

Definition

The natural exponential function uses Euler's number $e$ as the base: $f(x) = e^{x}$.

• Domain: $(-\infty, \infty)$

• Range: $(0, \infty)$

Summary Table: Exponential vs. Logarithmic Functions

Function Type	General Form	Domain	Range	Key Property
Exponential	$f(x) = a^{x}$	$(-\infty, \infty)$	$(0, \infty)$	Inverse of logarithmic
Logarithmic	$f(x) = \log_{a}(x)$	$(0, \infty)$	$(-\infty, \infty)$	Inverse of exponential