3-3 : Functions and their graphs Lesson objectives Students will be able to convert equations between logarithmic form and exponential form, evaluate common and natural logarithms, and graph common and natural logarithmic functions.

Warm Up Solve each equation. 8 = x3 2 x1/4 = 2 16 27 = 3x 3 46 = 43x

Quick Review

Quick Review Solutions

What you’ll learn about Inverses of Exponential Functions Common Logarithms – Base 10 Natural Logarithms – Base e Graphs of Logarithmic Functions … and why Logarithmic functions are used in many applications, including the measurement of the relative intensity of sounds.

Key Concepts : 𝑓 𝑥 = log 𝑏 𝑥 Logarithm- has base b of a positive number y is defined as follows: If 𝑦 = 𝑏 𝑥 , then log 𝑏 𝑦 = x. Common logarithm- a logarithm that uses base 10. ex. log 8

Logarithmic Functions are Inverses of Exponential Functions If 𝑎>0 and 𝑏>0, 𝑏≠ 1, then 𝑦= 𝑙𝑜𝑔 𝑏 𝑥 if and only if 𝑏 𝑦 =𝑥. Graph: 𝑦 = 𝑏 𝑥 𝑦= 𝑙𝑜𝑔 𝑏 𝑥 𝑦 = 𝑥

A. 𝑙𝑜𝑔 2 8 = 3 because B. 𝑙𝑜𝑔 3 3 = 1 2 because D. 𝑙𝑜𝑔 7 7 = 1 because Evaluating Logarithmic and Exponential Expressions A. 𝑙𝑜𝑔 2 8 = 3 because B. 𝑙𝑜𝑔 3 3 = 1 2 because C. 𝒍𝒐𝒈 𝟓 𝟏 𝟐𝟓 = -2 because D. 𝑙𝑜𝑔 7 7 = 1 because E. 6 𝑙𝑜𝑔 6 11 = 11 because

Logarithmic functions are inverses of exponential functions (x & y are switched) 𝑥 𝑓 𝑥 = 2 𝑥 𝑓 𝑥 = 𝑙𝑜𝑔 2 x -2 -1 1 2

Common Logarithms, Base 10 Logarithms with base 10 are called common logarithms. **The subscript 10 is often dropped, so a log statement with no specified base is understood to be base 10. EX #2: Evaluate the following logarithms and exponential expressions. A. log 100 C. log 1 1000 B. log 10 D. 10 log 6 6

Ex #3: Evaluate these common logarithms with a calculator. A. log 34.5 B. log 0.43 C. log (−3)

Solving Simple Logarithmic Equations To solve an exponential equation, change it to a logarithmic equation. To solve a logarithmic equation, change it to an exponential equation. Ex #4: Solve each equation by changing it to exponential form. A. log 𝑥=3 B. log 2 𝑥 =5

Natural Logarithms, Base e Notation: The logarithmic function 𝐥𝐨𝐠 𝒆 𝒙= 𝐥𝐧 𝒙 . EX #5: Evaluate the following logarithmic and exponential expressions. ln 𝑒 ln 𝑒 5 C. 𝑒 ln 4

Evaluating Natural Logarithms with a Calculator EX #6: Use a calculator to evaluate the logarithmic expressions. ln 23.5 = B. ln 0.48 C. ln −5 =

Independent Practice 3-3 : 2 to 36 even # - 20 minute

Key Concepts Logarithmic function- the inverse of an exponential function.

EX #6: Graph y = log4 x By definition of logarithm, y = log4 x is the inverse of y = 4x. Step 1: Graph y = 4x Step 2: Draw y = x. Step 3: Choose a few points on 4x. Reverse the coordinates and plot the points of y = log4 x.

Ex# 7 : Graph y = log5 (x – 1) + 2. Step 1: Graph y = log5 x Step 2: Graph the function by shifting the points from the graph to the right 1 unit and up 2 units.

Graphing Logarithmic Functions Basic function: 𝑓 𝑥 = 𝑙𝑛𝑥 or log 𝑥 Graph: Analysis:

EX #8: Transforming Logarithmic Graphs Describe how to transform the graph of 𝑦= ln𝑥 or 𝑦= log𝑥 into the graph of the given function. 𝐴.𝑔 𝑥 =ln⁡(𝑥+2) 𝐵.ℎ 𝑥 =ln⁡(3−𝑥) 𝑐.𝑔 𝑥 = 3log 𝑥 D. ℎ 𝑥 =1+ log 𝑥

Independent Practice 3-3 : 38 to 56 even # - 30 minute

Home work 3-3 : 63- 68 all Review 3-3 notes do all problems again.