Exponent Rules

Exponent Rules

• am * an = am+n, Demonstrated with x³ * x² = x5 = x3+2
• am / an = am-n (if a≠0), Demonstrated with x5 / x² = x3 = x5-2
• a0 = 1 (if a≠0), Demonstrated with x³/x³ = x0 = 1.
• (am)n = amxn, Demonstrated with (x3)2 = x6 = x3x2
• (ab)n = anbn, Demonstrated with (xy)² = x²y²
• (a/b)n = an/bn (if b≠0), Demonstrated with (x/2)³ = x³/2³

Additional Rules

• a-n = 1/an.
• am/n = n√am = (n√a)m

Rule Justification Examples

• x² / x5 = 1/x³ based on properties
• x2/5 = x-3 which equals 1/x³

Negative Exponent and Denominator Shortcut

• Flip the term to the other part of the fraction (numerator or denominator) and make the exponent positive.

Simplifying Expressions

• The goal is to rewrite expressions without negative or fractional exponents.

Examples of Simplification

• 2-3 = 1/23 = 1/8
• 3-4 = 1/34 = 1/81
• 5x-2 = 5 * (1/x²) = 5/x²
• 1001/2 = √100 = 10
• 82/3 = (3√8)² = 4
• 811/4 = 4√81 = 3
• 16-3/4 = 1/163/4 = 1/8
• 3-2 / (1/3²) = 9.

Combining Exponent Rules

• Simplify the given expression without negative or fractional exponents

Logarithms

• Logarithms are a new language.
• The goal is to translate "logs" into a more familiar language (exponents).

Translating Between Logs and Exponents

• logb A= x is equivalent to bx= A
• log form is the new language and exponential form is the old language.

Logarithm Terminology

• logb A is read as "log base b of A"
• b is the base of the logarithm.
• The base of a log can be any positive number except 1.

Translating Examples

• log2 8 = 3 → 23 = 8
• log36 (1/36) = -2 → 6-2 = 1/36
• log3 81 = 4 → 34 = 81

Translating with Circles

• Logs are actually exponents.
• log7 (1/49) = -2 → 7-2 = 1/49
• log2 32 = 5 → 25 = 32
• logD Q = P → DP = Q

Evaluating Logs

• To evaluate a log, ask what exponent is needed.
• The result of a logarithm is an exponent.

Evaluating Examples

• log7 49 = 2 because 72 = 49
• log3 81 = 4 because 34 = 81
• log2 (1/8) = -3 because 2-3 = 1/8

Common Log

• Base 10: log A = log10 A (base 10 is implied if no base is written).

Natural Log

• Base e: ln A = loge A (use "ln" instead of "log").

Calculator Usage

• Use a scientific or graphing calculator to evaluate common logs.
• Enter logs backwards on some calculators: 1000 log.

Log96 Evaluation

• log96 ≈ 1.9823
• 101.9823 ≈ 96

Natural Log

• e is an irrational number ≈ 2.7182818285

Evaluate In13

• ln13 ≈ 2.5649
• e2.5649 ≈ 13

Evaluate Ine8

• lne8 = 8

Change-of-Base Formula

• logb a = (loga) / (logb) = (lna) / (lnb)

Exponential Equations

• Variable appears in an exponent e.g., 2x = 5 or 34x+2 = 10.

Solving Exponential Equations

• An exponential equation is distinct from a quadratic equation
• Isolate the exponential part
• Translate to log form.
• Solve for the variable.

Sample Exponential Equation

• 5x= 11
• Exact answer: x = log5 11
• Approximate answer: x ≈ 1.4899
• 13 = 4 * 3x+5 + 5 can be solved by isolating 3x+5 first.
• Exact answer involves log3 2 - 5
• Approximate answer ≈ -4.3691

Exponential Growth

• Models exponential growth problems.

Sample Equation

• B = 1000e0.27t models the number of sulfur-oxidizing bacteria in a culture
• To find when B=1,000,000, solve for t
• t = ln(1,000) / 0.27≈ 26 hours.