Exponential and Logarithmic Functions Study Guide

Exponential and Logarithmic Functions

Overview

  • Two critical functions in mathematics and applications:

    • Exponential function: ( f(x) = a^x )

    • Logarithmic function: ( g(x) = \log_a x )

  • Applications: Population growth, radioactive decay, compound interest calculation.

Exponents

  • Definition: An exponent indicates how many times a number, known as the base, is multiplied by itself. For example, ( 2^5 = 2 \times 2 \times 2 \times 2 \times 2 ).

  • Base: The number being multiplied (e.g., 2 in ( 2^5 )).

  • Exponent: The number of times the base is used as a factor (e.g., 5 in ( 2^5 )).

  • Special Cases:

    • ( b^2 ): "b squared"

    • ( b^3 ): "b cubed"

  • Evaluation Examples:

    • ( 2^3 = 8 )

    • ( (-2)^3 = -8 )

    • ( -2^3 = - (2^3) = -8 )

Properties of Exponents

  • For any real number ( a ) and natural numbers ( m ) and ( n ):

    1. Multiplying powers: ( a^m \times a^n = a^{m+n} )

    2. Dividing powers: ( \frac{a^m}{a^n} = a^{m-n}, a \neq 0 )

    3. Power of a power: ( (a^m)^n = a^{mn} )

    4. Power of a product: ( (ab)^n = a^n b^n )

    5. Power of a quotient: ( \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}, b \neq 0 )

    6. Zero exponent: For any non-zero a, ( a^0 = 1 )

    7. Negative exponent: ( a^{-n} = \frac{1}{a^n}, a \neq 0 )

Logarithmic Functions

  • Definition: ( \log_a x = y ) means ( x = a^y )

  • Properties of Logarithms:

    1. Product Law: ( \loga(xy) = \loga x + \log_a y )

    2. Quotient Law: ( \loga \left(\frac{x}{y}\right) = \loga x - \log_a y )

    3. Power Law: ( \loga(x^r) = r \loga x )

    4. Change of Base Formula: ( \loga x = \frac{\logb x}{\log_b a} )

    5. Basic results: ( \loga a = 1 ) and ( \loga 1 = 0 )

Graphs of Exponential and Logarithmic Functions

  • Exponential function ( f(x) = a^x, a > 0, a \neq 1 )

    • Domain: All real numbers

    • Range: All positive real numbers

    • Asymptote: ( y=0 )

    • Characteristics: Rapid growth if ( a > 1 ); rapid decay if ( 0 < a < 1.

  • Logarithmic function ( g(x) = \log_a x )

    • Domain: All positive real numbers

    • Range: All real numbers

    • Asymptote: ( x=0 )

    • Characteristics: Increasing function.

Applications

  • Logarithms in Calculating pH:
    ( pH = -\log[H^+] )

  • Compound Interest Formula:

    • Compounded annually: ( A(t) = P(1 + r)^t )

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

  • Continuous Compounding:

    • Formula: ( A(t) = Pe^{rt} )

Exponential Growth Model

  • Population Growth:

    • Model: ( P(t) = P0 e^{kt} ), where ( P0 ) is initial population, and ( k ) is the growth rate.

    • Example: If a population starts at 110 million and grows at 2.3% per year, the growth function is ( P(t) = 110,000,000 e^{0.023t} ).

Review Exercises

  • Evaluate the following:

    1. ( 2^3 )

    2. ( \log_2(8) )

    3. Solve ( 5^x = 125 )

  • Convert between exponential and logarithmic forms:

    1. ( \log_2(4) = 2 ) translates to ( 2^2 = 4 )

    2. ( 3^{-1} = \frac{1}{3} )

Key Definitions

  • Exponent: A mathematical notation indicating the number of times to use the base in a multiplication.

  • Base: The number that is raised to a power.

  • Logarithm: The inverse operation to exponentiation, relating the base and its exponent to yield the result.

  • Exponential Function: A function that grows rapidly relative to its current value, typically represented as ( f(x) = a^x ).

  • Logarithmic Function: A function in the form ( g(x) = \log_a x ), where ( x ) is determined based on the output.