Logarithmic Functions Objectives Recognize and evaluate logarithmic functions with base a. Graph logarithmic functions with base a. Recognize, evaluate, and graph logarithmic functions with base e.

Definition: Logarithmic Function For x  0 and 0  a  1, y = loga x if and only if x = a y. The function given by f (x) = loga x is called the logarithmic function with base a. Every logarithmic equation has an equivalent exponential form: y = loga x is equivalent to x = a y A logarithm is an exponent! A logarithmic function is the inverse function of an exponential function. Exponential function: y = ax Logarithmic function: y = logax is equivalent to x = ay

Convert to exponential form: The logarithmic function to the base a, where a > 0 and a  1 is defined: y = logax if and only if x = a y logarithmic form exponential form When you convert an exponential to log form, notice that the exponent in the exponential becomes what the log is equal to. Convert to log form: Convert to exponential form:

LOGS = EXPONENTS With this in mind, we can answer questions about the log: This is asking for an exponent. What exponent do you put on the base of 2 to get 16? (2 to the what is 16?) What exponent do you put on the base of 3 to get 1/9? (hint: think negative) What exponent do you put on the base of 4 to get 1? When working with logs, re-write any radicals as rational exponents. What exponent do you put on the base of 3 to get 3 to the 1/2? (hint: think rational)

Examples: Write Equivalent Equations Examples: Write the equivalent exponential equation and solve for y. Solution Equivalent Exponential Equation Logarithmic Equation y = log216 16 = 2y 16 = 24  y = 4 y = log2( ) = 2 y = 2-1 y = –1 y = log416 16 = 4y 16 = 42  y = 2 y = log51 1 = 5 y 1 = 50  y = 0

Your Turn: Write each equation in exponential form log 3 81 = 4 34=81 log 7 1/49 = -2 7-2=1/49 Write each equation in logarithmic form 103 = 1000 Log101000=3 4-2 = 1/16 Log41/16=-2

Your Turn: 25 Find y in each equation. log 2 8 = y log 5 1 = y y=3

Properties of Logarithms 1. loga 1 = 0 since a0 = 1. 2. loga a = 1 since a1 = a. 3. loga ax = x and alogax = x inverse property 4. If loga x = loga y, then x = y. one-to-one property Examples: Solve for x: log6 6 = x log6 6 = 1 property 2 x = 1 Simplify: log3 35 Simplify: 7log79 log3 35 = 5 property 3 7log79 = 9 property 3

Your Turn: Solve 1 30

Base 10 logarithms Called common logarithms When base a is not indicated, it is understood that a = 10 log 1/100 = log 10 = log 1/10 = log 100 = log 1 = log 1000 = The LOG key on your calculator. -2 1 -1 2 3

In the last section we learned about the graphs of exponentials. Logs and exponentials are inverse functions of each other so let’s see what we can tell about the graphs of logs based on what we learned about the graphs of exponentials. Recall that for functions and their inverses, x’s and y’s trade places. So anything that was true about x’s or the domain of a function, will be true about y’s or the range of the inverse function and vice versa. Let’s look at the characteristics of the graphs of exponentials then and see what this tells us about the graphs of their inverse functions which are logarithms.

1. Domain is all real numbers 1. Range is all real numbers Characteristics about the Graph of an Exponential Function a > 1 Characteristics about the Graph of a Log Function a > 1 1. Domain is all real numbers 1. Range is all real numbers 2. Range is positive real numbers 2. Domain is positive real numbers 3. There are no x intercepts because there is no x value that you can put in the function to make it = 0 3. There are no y intercepts 4. The x intercept is always (1,0) (x’s and y’s trade places) 4. The y intercept is always (0,1) because a 0 = 1 5. The graph is always increasing 5. The graph is always increasing 6. The x-axis (where y = 0) is a horizontal asymptote for x  -  6. The y-axis (where x = 0) is a vertical asymptote

base a>1 Logarithmic Graph Exponential Graph Graphs of inverse functions are reflected about the line y = x

Graph f(x) = log2 x Since the logarithm function is the inverse of the exponential function of the same base, its graph is the reflection of the exponential function in the line y = x. x y y = 2x y = x 8 3 4 2 1 –1 –2 2x x horizontal asymptote y = 0 y = log2 x x-intercept (1, 0) vertical asymptote x = 0

Graphs of Logarithmic Functions The graphs of logarithmic functions are similar for different values of a. f(x) = loga x (a  1) y-axis vertical asymptote x y Graph of f (x) = loga x (a  1) y = a x y = x range 1. domain y = log2 x 2. range 3. x-intercept (1, 0) 4. vertical asymptote domain x-intercept (1, 0) 5. increasing 6. continuous 7. one-to-one 8. reflection of y = a x in y = x

Graphs of Logarithmic Functions Typical shape for graphs where a > 1 (includes base e and base 10 graphs). Typical shape for graphs where 0 < a < 1.

The Logarithmic Function: f (x) = loga x, a > 1

The Logarithmic Function: f (x) = loga x, 0 < a < 1

Determining Domains of Logarithmic Functions Example Find the domain of each function. Solution Argument of the logarithm must be positive. x – 1 > 0, or x > 1. The domain is (1,). Use the sign graph to solve x2 – 4 > 0. The domain is (–,–2) (2, ).

Domain: (-3, 3) or -3<x <3 Your Turn: Find the domain. (a) Solution Domain: (-3,) or x >-3 Domain: (-3, 3) or -3<x <3

Natural Logarithms Any log to the base e is known as a In French this is a logarithme naturel Which is where ln comes from. When you see ln (instead of log) then it’s a natural log y = ln x is the inverse of y = ex The LN key on your calculator.

Natural Logarithms

Natural Logarithms

Natural Logarithms

Natural Logarithms