C4

Exponential and Logarithmic Functions

Exponential Functions

Exponential functions are mathematical expressions in which a constant base is raised to a variable exponent. They are commonly used to model growth and decay in various contexts.

• General Form: $y = ab^x$

• Example: Find the equation of the exponential curve $y = ab^x$ that passes through the points (3, -1) and (1, -16).

Additional info: Exponential functions are the inverse of logarithmic functions.

Logarithmic Functions

A logarithmic function is the inverse of an exponential function. It answers the question: "To what exponent must the base be raised to produce a given number?"

• General Form: $f(x) = ext{log}_b x$ for $x > 0$ and $b > 0$, $b \neq 1$

• Definition: $y = ext{log}_b x$ is equivalent to $b^y = x$

• Example: $y = ext{log}_2 8$ is equivalent to $2^y = 8$

Converting Between Exponential and Logarithmic Forms

Conversion Table

Logarithmic and exponential forms are interchangeable. The following table summarizes common conversions:

Logarithmic Form	Exponential Form	Answer
$ext{log}_{10} 100 = x$	$10^x = 100$	$x = 2$
$ext{log}_2 8 = x$	$2^x = 8$	$x = 3$
$ext{log}_5 25 = x$	$5^x = 25$	$x = 2$
$ext{log}_3 1 = x$	$3^x = 1$	$x = 0$
$ext{log}_2 rac{1}{8} = x$	$2^x = rac{1}{8}$	$x = -3$
$ext{log}_4 16 = x$	$4^x = 16$	$x = 2$
$ext{log}_2 x = 5$	$2^5 = x$	$x = 32$
$ext{log}_x 25 = 2$	$x^2 = 25$	$x = 5$

Evaluating Logarithms Without a Calculator

Examples

• $ext{log}_{10} 100 = 2$

• $ext{log}_2 8 = 3$

• $ext{log}_5 25 = 2$

• $ext{log}_3 1 = 0$

• $ext{log}_2 rac{1}{8} = -3$

Properties of Logarithms

Basic Properties

• Logarithm of One: $ext{log}_b 1 = 0$

• Inverse Property: $ext{log}_b b = 1$

• Inverse (Think "undo"): $ext{log}_b b^x = x$ and $b^{ ext{log}_b x} = x$

Product, Quotient, and Power Rules

Rule	Formula
Product Rule	$ext{log}_b (xy) = ext{log}_b x + ext{log}_b y$
Quotient Rule	$ext{log}_b igg(\frac{x}{y}\bigg) = ext{log}_b x - ext{log}_b y$
Power Rule	$ext{log}_b (x^r) = r ext{log}_b x$

Condensing Logarithmic Expressions

Use the properties above to write a sum or difference of logarithms as a single logarithm.

• Example: $ext{log}_2 x + ext{log}_2 y = ext{log}_2 (xy)$

• Example: $ext{log}_2 x - ext{log}_2 y = ext{log}_2 \left(\frac{x}{y}\right)$

• Example: $2 ext{log}_2 x = ext{log}_2 x^2$

Common Logarithms and Natural Logarithms

Common Logarithm

• $ext{log}_{10} x$ is usually written as $ext{log} x$

Natural Logarithm

• $ext{log}_e x$ is usually written as $ext{ln} x$

• $e$ is an irrational number, approximately $2.71828$

Change of Base Property

The change of base formula allows you to rewrite logarithms in terms of logs with a different base, which is useful for calculation.

• Formula: $ext{log}_b x = \frac{ ext{log}_a x}{ ext{log}_a b}$

• Example: $ext{log}_2 5 = \frac{ ext{log}_{10} 5}{ ext{log}_{10} 2}$

Solving Exponential Equations Using Logarithms

Logarithms are useful for solving equations where the variable is in the exponent. By taking the logarithm of both sides, you can "bring the exponent down" and solve for the variable.

• One-to-One Property: If $b^M = b^N$, then $M = N$

• Example: Solve $2^x = 50$:

  • Take log of both sides: $ext{log} 2^x = ext{log} 50$

  • Apply power rule: $x ext{log} 2 = ext{log} 50$

  • Solve for $x$: $x = \frac{ ext{log} 50}{ ext{log} 2} \approx 5.644$

Summary Table: Properties of Logarithms

Property	Formula
Definition	$y = ext{log}_b x \iff b^y = x$
Inverse Properties	$ext{log}_b b^x = x$ $b^{ ext{log}_b x} = x$
Log Property Involving One	$ext{log}_b 1 = 0$
Product Rule	$ext{log}_b (xy) = ext{log}_b x + ext{log}_b y$
Quotient Rule	$ext{log}_b \left(\frac{x}{y}\right) = ext{log}_b x - ext{log}_b y$
Power Rule	$ext{log}_b (x^r) = r ext{log}_b x$
One-to-One Properties	If $b^M = b^N$, then $M = N$ If $ext{log}_b M = ext{log}_b N$, then $M = N$

Practice Problems

• Expand and condense logarithmic expressions using product, quotient, and power rules.

• Solve exponential equations by applying logarithms and their properties.

• Convert between logarithmic and exponential forms.

• Evaluate logarithms without a calculator using properties and known values.

Additional info: These notes cover the essential properties and applications of logarithms, including solving exponential equations, which are core topics in College Algebra.