Precalculus Review: Exponential and Logarithmic Functions, Inverses, and Properties of Exponents

Inverse Functions

Definition and Properties

Inverse functions are fundamental in mathematics, allowing us to 'reverse' the effect of a function. A function f has an inverse if and only if it is one-to-one (injective), meaning each output is produced by exactly one input.

• Definition: Functions f and g are inverses if $f(g(x)) = x$ and $g(f(x)) = x$ for all x in their domains.

• Graphical Interpretation: The graph of an inverse function is a reflection of the original function across the line $y = x$.

Example: If $f(x) = 2x + 3$, then its inverse is $f^{-1}(x) = \frac{x-3}{2}$.

Finding Inverses

• Replace $f(x)$ with $y$.

• Solve for $x$ in terms of $y$.

• Interchange $x$ and $y$; solve for $y$ to get $f^{-1}(x)$.

Example: Find the inverse of $f(x) = \frac{2x+5}{x-1}$, $x \neq 1$.

• Let $y = \frac{2x+5}{x-1}$

• Solve for $x$:

• $y(x-1) = 2x + 5$

• $yx - y = 2x + 5$

• $yx - 2x = y + 5$

• $x(y-2) = y + 5$

• $x = \frac{y+5}{y-2}$

• Interchange $x$ and $y$: $y = \frac{x+5}{x-2}$

• Thus, $f^{-1}(x) = \frac{x+5}{x-2}$

Example: Find the inverse of $f(x) = e^{x+1} + 5$.

• Let $y = e^{x+1} + 5$

• $y - 5 = e^{x+1}$

• $\ln(y-5) = x+1$

• $x = \ln(y-5) - 1$

• Interchange $x$ and $y$: $y = \ln(x-5) - 1$

• Thus, $f^{-1}(x) = \ln(x-5) - 1$

Properties of Exponents

Exponent Rules

Exponents follow several key properties that simplify expressions and solve equations.

• Product of Powers: $a^m \cdot a^n = a^{m+n}$

• Quotient of Powers: $\frac{a^m}{a^n} = a^{m-n}$

• Power of a Power: $(a^m)^n = a^{mn}$

• Power of a Product: $(ab)^n = a^n b^n$

• Power of a Quotient: $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$

Rational Exponents

Rational exponents represent roots and powers in a unified way.

• $a^{1/n} = \sqrt[n]{a}$

• $a^{m/n} = (\sqrt[n]{a})^m = \sqrt[n]{a^m}$

Example: $\left(\frac{1}{25}\right)^{3/2} = (25^{-1})^{3/2} = 25^{-3/2} = \frac{1}{25^{3/2}} = \frac{1}{(\sqrt{25})^3} = \frac{1}{5^3} = \frac{1}{125}$

Exponential Functions

Definition and Types

An exponential function has the form $f(x) = a^x$, where $a > 0$ and $a \neq 1$.

• If $a > 1$, the function models exponential growth (increasing function).

• If $0 < a < 1$, the function models exponential decay (decreasing function).

Graphs of Exponential Functions

• The domain is $(-\infty, \infty)$.

• The range is $(0, \infty)$.

• The horizontal asymptote is $y = 0$.

Example Table: For $f(x) = 2^x$:

x	$2^x$
-2	1/4
-1	1/2
0	1
1	2
2	4
3	8
4	16

Logarithmic Functions

Definition and Relationship to Exponentials

A logarithmic function is the inverse of an exponential function. For $a > 0$, $a \neq 1$:

• $y = \log_a x$ if and only if $a^y = x$

Logarithmic Form: $y = \log_a x$

Exponential Form: $a^y = x$

Graphing Logarithms

• The domain is $(0, \infty)$.

• The range is $(-\infty, \infty)$.

• The vertical asymptote is $x = 0$.

Example Table: For $g(x) = \log_2 x$:

x	$\log_2 x$
1/2	-1
1	0
2	1
4	2
8	3

Properties of Logarithms

• Product Rule: $\log_a(xy) = \log_a x + \log_a y$

• Quotient Rule: $\log_a\left(\frac{x}{y}\right) = \log_a x - \log_a y$

• Power Rule: $\log_a(x^r) = r \log_a x$

• Logarithm of 1: $\log_a 1 = 0$

• Logarithm of the Base: $\log_a a = 1$

• Change of Base Formula: $\log_a x = \frac{\log_b x}{\log_b a} = \frac{\ln x}{\ln a}$

Applications of Exponential and Logarithmic Functions

Exponential Growth and Decay

Many real-world phenomena, such as population growth and radioactive decay, are modeled by exponential functions.

• General Exponential Model: $y = y_0 e^{kt}$

• If $k > 0$, the process is growth; if $k < 0$, it is decay.

Half-life: The time required for a quantity to reduce to half its initial value. For decay constant $k$:

• $t_{1/2} = \frac{\ln 2}{|k|}$

Compound Interest

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

• Continuously compounded: $A = Pe^{rt}$

pH and Logarithms

The pH of a solution is a logarithmic measure of hydronium ion concentration:

• $\mathrm{pH} = -\log_{10}[\mathrm{H}_3\mathrm{O}^+]$

Example: If pH = 7.68, then $[\mathrm{H}_3\mathrm{O}^+] = 10^{-7.68}$ M.

Summary Table: Exponential and Logarithmic Properties

Property	Exponential	Logarithmic
Product	$a^m \cdot a^n = a^{m+n}$	$\log_a(xy) = \log_a x + \log_a y$
Quotient	$\frac{a^m}{a^n} = a^{m-n}$	$\log_a\left(\frac{x}{y}\right) = \log_a x - \log_a y$
Power	$(a^m)^n = a^{mn}$	$\log_a(x^r) = r \log_a x$
Change of Base	N/A	$\log_a x = \frac{\log_b x}{\log_b a}$

Additional info: These notes summarize key Precalculus concepts on inverses, exponents, exponentials, logarithms, and their applications, suitable for exam review.