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. The logarithm of a number is the exponent to which the base must be raised to produce that number.

• General Form: $y = ext{log}_b x$

• Equivalent Exponential Form: $b^y = x$

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

Interpreting Logarithmic Statements

Converting Between Logarithmic and Exponential Forms

Logarithmic equations can be rewritten in exponential form and vice versa. This is useful for solving equations and understanding the relationship between the two forms.

• Example Table:

Logarithmic Form	Exponential Form	Answer
$ext{log}_{10} x = 2$	$10^2 = x$	$x = 100$
$ext{log}_2 x = 5$	$2^5 = x$	$x = 32$
$ext{log}_3 x = -2$	$3^{-2} = x$	$x = \frac{1}{9}$
$ext{log}_x 25 = 2$	$x^2 = 25$	$x = 5$ or $x = -5$
$ext{log}_x 8 = 3$	$x^3 = 8$	$x = 2$
$ext{log}_2 x = -5$	$2^{-5} = x$	$x = \frac{1}{32}$
$ext{log}_x 81 = 4$	$x^4 = 81$	$x = 3$
$ext{log}_x 1 = 0$	$x^0 = 1$	$x \neq 0$

Evaluating Logarithms Without a Calculator

Common Logarithms and Their Values

Some logarithms can be evaluated "mentally" by recognizing powers of the base.

• Examples:

• $ext{log}_{10} 100 = 2$ because $10^2 = 100$

• $ext{log}_2 8 = 3$ because $2^3 = 8$

• $ext{log}_5 25 = 2$ because $5^2 = 25$

• $ext{log}_3 1 = 0$ because $3^0 = 1$

Properties of Logarithms

Basic Properties

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

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

• Inverse Relationship: $ext{log}_b b^x = x$ 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	$ext{log}_b \left(\frac{x}{y}\right) = ext{log}_b x - ext{log}_b y$
Power Rule	$ext{log}_b (x^r) = r ext{log}_b x$

Condensing Logarithmic Expressions

Multiple logarithmic terms can be combined into a single logarithm using the above rules.

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

• Example: $2 ext{log}_b x - ext{log}_b y = ext{log}_b (x^2/y)$

Expanding Logarithmic Expressions

Logarithmic expressions can be expanded using the product, quotient, and power rules.

• Example: $ext{log}_b (xy^2) = ext{log}_b x + 2 ext{log}_b y$

• Example: $ext{log}_b \left(\frac{x^3}{y}\right) = 3 ext{log}_b x - ext{log}_b y$

Common and Natural Logarithms

Common Logarithm

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

Natural Logarithm

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

Change of Base Property

The change of base formula allows you to rewrite logarithms in terms of logs with a different base.

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

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

Solving Exponential and Logarithmic Equations

Solving Exponential Equations Using Logarithms

Logarithms are useful for solving equations where the variable is in the exponent. The one-to-one property of logarithms allows us to set exponents equal when the bases are the same.

• Example: Solve $3^x = 50$

• Solution: $x = \text{log}_3 50$

• Decimal Approximation: $x \approx 3.560$

One-to-One Properties

• If $b^M = b^N$, then $M = N$

• If $ext{log}_b M = ext{log}_b N$, then $M = N$

Summary Table: Properties of Logarithms

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

Practice Problems and Applications

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

• Solve exponential equations by taking logarithms of both sides.

• Apply the change of base formula to evaluate logarithms with arbitrary bases.

• Evaluate logarithms without a calculator by recognizing powers of the base.

Additional info: Mastery of logarithmic properties is essential for solving equations and simplifying expressions in algebra and calculus.