Download presentation
Presentation is loading. Please wait.
1
Logarithms and Logarithmic Functions
Section 6.3 Beginning on page 310
2
Logarithms log π π¦ =π₯ π π₯ =π¦ π π₯ =π¦ log π π¦ =π₯
For what value of x does 2 π₯ =6 ? Logarithms can answer this question. Log is the inverse operation to undo unknown exponents. log π π¦ =π₯ π π₯ =π¦ π π₯ =π¦ log π π¦ =π₯ *Read as log base b of y Exponential form Logarithmic form
3
Rewriting Logarithmic Equations
Example 1: Rewrite each equation in exponential form. log =4 log 4 1 =0 log =1 log =-1 2 4 =16 4 0 =1 12 1 =12 1 4 β1 =4 log π π¦ =π₯ π π₯ =π¦
4
Rewriting Exponential Functions
Example 2: Rewrite each equation logarithmic form. 5 2 =25 10 β1 =0.1 =4 6 β3 = 1 216 log =2 log =β1 log 8 4 = 2 3 log =β3 π π₯ =π¦ log π π¦ =π₯
5
Evaluating Logarithmic Expressions
Example 3: Evaluate each logarithm a) log b) log c) log d) log 36 6 1 5 ? =125 4 ? =64 5 ? =0.2 36 ? =6 5 ? = 1 5 1 2 4 -3 β1
6
Evaluating Common and Natural Logs
A common logarithm is logarithm with base 10. It is denoted by log or simply log. A natural logarithm is a logarithm with base π and is usually denoted by ln. Example 4: Evaluate (a) log 8 and (b) ln 0.3 using a calculator. Round your answer to three decimal places. a) log 8β0.903 b) ln 0.3 ββ1.204
7
Using Inverse Properties
Exponential functions and logarithmic functions undo each other: log π π π₯ =π₯ and π log π π₯ =π₯ Example 5: Simplify a) 10 log b) log π₯ =4 = log π₯ = log π₯ =2π₯
8
Finding Inverse Functions
Example 6: Find the inverse of each function. a) π π₯ = 6 π₯ b) π¦= ln (π₯+3) π₯= 6 π¦ π₯= ln (π¦+3) π¦= log 6 π₯ π¦+3= π π₯ f(π₯)= log 6 π₯ π¦= π π₯ β3
9
Graphing Logarithmic Functions
JUST SKETCH GRAPHS
10
Graphing a Logarithmic Function
Example 7: Graph π π₯ = log 3 π₯ Find the inverse function. Create a table of values for π π₯ Swap the values of x and y to create a table for π π₯ . 4) Plot the points and connect them with a smooth curve. π₯= log 3 π¦ π¦= 3 π₯ π(π₯)= 3 π₯ x -2 -1 1 2 G(x) 1 9 1 3 1 3 9 X π π π π 1 3 9 F(x) -2 -1 2
11
Monitoring Progress Rewrite the equation in exponential form.
1) log =4 2) log 7 7 =1 3) log =0 4) log =β5 Rewrite the equation in logarithmic form. 5) 7 2 =49 6) =1 7) 4 β1 = ) =2 Evaluate the logarithm. If necessary, use a calculator and round your answer to three decimal places. 9) log ) log ) log ) ln 0.75 3 4 =81 7 1 =7 1 2 β5 =32 14 0 =1 log =2 log =0 log =β1 log = 1 8 = 1 3 =5 β1.079 ββ0.288
12
Monitoring Progress 13) 8 log 8 π₯ 14) log 7 7 β3π₯
Simplify the expression: 13) 8 log 8 π₯ ) log β3π₯ 15) log π₯ ) π ln 20 17) Find the inverse of π¦= 4 π₯ 18) Find the inverse of π¦= ln (π₯β5) =π₯ =β3π₯ =6π₯ =20 π¦= log 4 π₯ π¦= π π₯ +5
13
Monitoring Progress Graph the function.
19) π¦= log 2 π₯ ) π π₯ = log 5 π₯ ) π¦= log π₯
Similar presentations
© 2025 SlidePlayer.com Inc.
All rights reserved.