Logarithms and Logarithmic Functions Unit 6.6

Warm up… The Richter scale describes the intensity of an earthquake. It was developed by Charles Richter in 1935. The table shows how the intensity of an earthquake increases as the number increases. Use a graphing calculator to make a scatter plot for the data in the table. Sketch the graph in your notes including the window. 2. Find an exponential equation to fit this data.

Warm up… 1. 2.

Reflecting Exponential Functions Work and complete the worksheet when you have finished your quiz.

How do I solve x = 3y ? John Napier was a Scottish theologian and mathematician who lived between 1550 and 1617. He spent his entire life seeking knowledge, and working to devise better ways of doing everything from growing crops to performing mathematical calculations. He invented a new procedure for making calculations with exponents easier by using what he called logarithms. A logarithm can be written as a function y = logbx. The notation y = logbx is another way of writing x = by. So x = by and y = logbx represent the same functions. y =log3x is simply another way of writing x = 3y. The notation is read “y is equal to the logarithm, base 3, of x.”

Logarithmic Function (Common) Calculator: y1= log(x) Domain: x > 0 or x  (0,∞) Range: y   or y  (-∞,∞) Zeros: (1,0) or x = 1 X-Intercept: (1,0) Y-Intercept: none

Logarithmic Function (Common) Symmetry: None Max: None Min: None Increasing: x  (0,∞) Decreasing: Never Vertical Asymptotes: x = 0 Horizontal Asymptotes: None

Logarithmic Function (Natural) Calculator: y1= ln(x) Domain: x > 0 or x  (0,∞) Range: y   or y  (-∞,∞) Zeros: (1,0) or x = 1 X-Intercept: (1,0) Y-Intercept: none

Logarithmic Function (Natural) Symmetry: None Max: None Min: None Increasing: x  (0,∞) Decreasing: Never Vertical Asymptotes: x = 0 Horizontal Asymptotes: None

Things I should Know about Logarithmic Functions The output of a log function is an exponent. Log and exp. functions are inverses. Domain: (0, ) Range: (-, ) x-intercept of the graph: (1,0) Vertical Asymptote at x = 0 (or the y-axis)

Definition of Logarithm Exponential Form Logarithmic Form (base)(exp) = (product) (exp) = log(base)(product) EX: 5y = 25 y = log525. THE QUESTION that you’re trying to answer: What exponent, y, takes a base of b to a product of x? EX: What exponent, y, takes a base of 5 to 25? y = 2

Example 1…. Change from Log notation to Exponential notation or Exponential notation to Log notation. Logarithmic Form Exponential Form (___ is the exp. that takes a base of __ to a __.) 2 3 9 (___ is the exp. that takes a base of __ to an __.) 3 x 8

Example 2…. Change from Log notation to Exponential notation or Exponential notation to Log notation. Logarithmic Form Exponential Form (__ is the exp. that takes a base of __ to an ___.) 4 3 81

Only One Graph Is Visible. Equivalent Equations x = 3y and y = log3x Only One Graph Is Visible.

Two graphs are visible that are reflected over the y = x line. Inverse Equations The inverse of the exponential parent function can be defined as a new function, the logarithmic parent function. The functions are reflections of each other over the line y = x. y = 2x and y = log2x Two graphs are visible that are reflected over the y = x line.

Finding the inverse algebraically Example 3: Rewrite the inverse of the exponential function y = 2x + 3 as a logarithmic function. Beginning Equation Replace x with y and y with x. Isolate the term containing y. Rewrite the exponential function as a logarithmic function

Finding the inverse remember… When using a table of values to find the inverse of an exponential function the domain will switch with the range like on the beginning activity. When using a graph to find the inverse of an exponential function, the graph must reflect over the y = x line. When using algebraic methods to find the inverse of an exponential function, switch x and y in the equation, get the term containing y by itself, and then rewrite in logarithmic form.

Example 4 y = 4x-2. log2x+3 x = 4y-2. x = log2y+3 x+2 = 4y x-3 = log2y Determine the inverse of the following. Your final answer should be in the equation box below. y = 4x-2. log2x+3 x = 4y-2. x = log2y+3 x+2 = 4y x-3 = log2y y=log(x+2) y=2x-3

Example 4 Determine the inverse of the following. y = 6x-5 y = log6(x+2) x = 6y-5 x = log6(y+2) log6x= y-5 6x = y+2 log6x+5=y Or y = log6x+5 y=6x - 2

Example 5…. Evaluate the expression. Set equal to x Re-write in exponential form. Solve for x.

Example 5… Definition of Logarithms

Example 6… Definition of Logarithms Property of Equality Equate the Exponents Solve for x

Cont… The solution checks!

Example 7… Definition of Logarithms Power of Power Equate the Exponents Check your solution!

Example 7… The solution checks!

Forms: Logarithmic Functions Parent Function: Common Log: Natural Log: Standard Form: Transfm Form:

Absent Students-Notes 6.6 Attach this note page into your notebook Complete all examples and warm-ups Be sure to understand the difference between finding the inverse, evaluating, solving and rewriting logs and exponentials