MAT122::Lecture Note::Week 10

Home	Previous	Next

Overview

Can this be true?
Quickie review of inverse functions.
Continue with logarithms and logarithmic functions.

Assignment(s):

Quote of the Week {Furman.edu:: Mathematical Quotations Server}
The future is something which everyone reaches at the rate
of sixty minutes an hour, whatever he does, whoever he is.
-- C.S. Lewis (01898-01963) {Irish author; more...} [future] [log]
BARS of the Week {NASA.gov:: Astronomy Picture of the Day}
Pueblo CO | Aurora CO | Brush CO [log]

Introduction to Inverse Functions

Some functions have inverse functions. A function has an inverse function if the inputs and outputs of the original function can be swapped such that they still represent a function (i.e. each input maps to one and only one output).

[definition] A function is a one-to-one function if each input value in the function's domain produces a different (unique) output value. The horizontal line test can be used to check if a graph of a function is a graph of a one-to-one function.

The inverse of function named f is f^-1. Note: The -1 is not an exponent--it is part of the name of the inverse function.

The following two tables are used to introduce the concept of inverse functions.
function f() ============ 0 1 2 3 4 <-- inputs --------------------- 5 3 1 3 5 <-- outputs Function f() does not have an inverse function because using the 2nd row as inputs cannot be a function. Input of 5 produces two different outputs (as does an input 3). function g() ============ 0 1 2 3 4 <-- inputs --------------------- 5 3 1 8 2 <-- outputs Function g() has an inverse because the inputs can be swapped with the outputs resulting in a function. g(0) = 5 and g^-1(5) = 0 g(1) = 3 and g^-1(3) = 1 g(2) = 1 and g^-1(1) = 2 g(3) = 8 and g^-1(8) = 3 g(4) = 2 and g^-1(2) = 4

If the output of a function is used as input to the function's inverse, then the inverse function outputs the original input.
if f(n) = a, then f^-1(a) = n assume f(n) = 2n + 1 let y = f(n) such that y = 2n + 1 solve for n to get n = (y - 1) / 2 f^-1(n) = (y - 1) / 2 f(5) = 2(5) + 1 = 11 f^-1(11) = (11 - 1) / 2 = 5 f(5) = 11, f^-1(11) = 5

The following algorithm can be used to find the inverse function of a one-to-one function.
   If written in f(x) notation, replace f(x) with y.
   Interchange the variables x and  y.
   Solve for y.
   Substitute f^-1(x) for  y.
The following is true with respect to function composition.
(f o f^-1)(x) = x ...or... f(f^-1(x)) = x

{TopOfPage} {101Science} {NameThatFunction} { TI-83} { Factors} {MathWords}

Introduction to Logarithms

Logarithmic functions are inverses of exponential functions. In a nutshell, logarithms are exponents.

Quickie review of exponent terminology...

   b^p  is the number 'b' raised-to-the-power 'p'
   'b' is called the base and 'p' is the exponent (power)
   [somebody ask me:  why 'p' instead of 'e'?]
   
   examples:

   5^3 = 125    [5 * 5 * 5]
   2^4 = 16     [2 * 2 * 2 * 2]
   5^0 = 1      [by definition]

A base-b logarithm of an input number n is the power p to which b must be raised-to in order for it to equal n.

log_b(n) = p implies b^p = n examples: log₂(8) = 3 implies 2^3 = 8 log₁₀(100) = 2 implies 10^2 = 100 log₇(7) = 1 implies 7^1 = 7 log₁₆(16^5) = 5 implies 16^5 = 1,048,576 log₅(625) = 4 implies 5^4 = 625

A base-10 logarithm is called a common logarithm and it is named log(n). In other words, if no base is specified, then it defaults to 10.

log(n) = p implies log₁₀(n) = p examples: log(10) = 1 implies 10^1 = 10 log(1000) = 3 implies 10^3 = 1000 log(10^9) = 9 implies 10^9 = 1,000,000,000 log(51) = 1.708 implies 10^1.708 ~= 51

A base-e logarithm is called a natural logarithm and it is typically named ln(n).

ln(n) = p equals log_e(n) = p which implies e^p = n examples: ln(51) = 3.932 implies e^3.932 ~= 51 ln(122) = 4.804 implies e^4.804 ~= 122 ln(e^3) = 3 implies e^3 ~= 20.0855 ln(e) = 1 implies e^1 = e

Many calculators have buttons for calculating common (base-10) and natural logarithms (base-e); however, they don't have buttons for calculating logarithms in other bases. The "change of base" formula can be used to switch logarithms from one base to another.

log_a(n) = log_b(n) / log_b(a) where 'a' is the given base and 'b' is the change-to base examples: (a) log₃(5) = log(5) / log(3) change from base-3 to base-10 (b) log₇(11) = log(11) / log(7) change from base-7 to base-10 (c) log₁₆(32) = log₂(32) / log₂(16) change from base-16 to base-2 (d) log₅(122) = ln(122) / ln(5) change from base-5 to base-e (a) log₃(5) = log(5) / log(3) = 0.699 / 0.477 = 1.464 observe... 3^1.464 ~= 5 (b) log₇(11) = log(11) / log(7) = 1.041 / 0.845 = 1.232 observe... 7^1.232 ~= 11 (c) log₁₆(32) = log₂(32) / log₂(16) = 5 / 4 = 1.25 observe... 16^1.25 = 32 (d) log₅(122) = ln(122) / ln(5) = 4.804 / 1.609 = 2.985 observe... 5^2.985 ~= 122

{TopOfPage} {101Science} {NameThatFunction} { TI-83} { Factors} {MathWords}

Logarithm Laws

The following logarithm laws hold true for all bases.

Assume n > 0 and m > 0 log(n) = log n ...sometimes ()'s are not used around the input variable log(n) = log₁₀(n) ...if no base is specified, then it defaults to 10 ...base-10 logs are called "common logs" log_e(n) = ln(n) ...base-e logs are called "natural logs" log(1) = 0 ...any number raised to the zero power is 1 log_b(n) = 1, when b=n ...any number raised to the first power is that number log_b(n^p) = p log(n * m) = log(n) + log(m) log(n / m) = log(n) - log(m) log(n^m) = m * log(n)

It is important to remember the following.

log(n + m) ≠ log(n) + log(m) log(n - m) ≠ log(n) - log(m) log(n * m) ≠ log(n) * log(m) log(n / m) ≠ log(n) / log(m)

The following are "change of base" formulas.

log_a(n) = log_b(a) / log_b(a) ...typically 'b' will be either 10 or e log_a(n) = ln(a) / ln(a)

Note: log(n) equals a complex number with a non-zero imaginary part when 'n' is negative.

Math.ASU.edu::MAT117::Laws of Logarithmic Functions

{TopOfPage} {101Science} {NameThatFunction} { TI-83} { Factors} {MathWords}

Home Previous Next