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Logarithms and Logarithmic Functions

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Presentation on theme: "Logarithms and Logarithmic Functions"β€” Presentation transcript:

1 Logarithms and Logarithmic Functions
Section 6.3 Beginning on page 310

2 Logarithms log 𝑏 𝑦 =π‘₯ 𝑏 π‘₯ =𝑦 𝑏 π‘₯ =𝑦 log 𝑏 𝑦 =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 𝑏 𝑦 =π‘₯ 𝑏 π‘₯ =𝑦 𝑏 π‘₯ =𝑦 log 𝑏 𝑦 =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 βˆ’1 =4 log 𝑏 𝑦 =π‘₯ 𝑏 π‘₯ =𝑦

4 Rewriting Exponential Functions
Example 2: Rewrite each equation logarithmic form. 5 2 =25 10 βˆ’1 =0.1 =4 6 βˆ’3 = 1 216 log =2 log =βˆ’1 log 8 4 = 2 3 log =βˆ’3 𝑏 π‘₯ =𝑦 log 𝑏 𝑦 =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 βˆ’1

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 𝑒 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 8β‰ˆ0.903 b) ln 0.3 β‰ˆβˆ’1.204

7 Using Inverse Properties
Exponential functions and logarithmic functions undo each other: log 𝑏 𝑏 π‘₯ =π‘₯ and 𝑏 log 𝑏 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) 𝑔 π‘₯ = 6 π‘₯ b) 𝑦= ln (π‘₯+3) π‘₯= 6 𝑦 π‘₯= ln (𝑦+3) 𝑦= log 6 π‘₯ 𝑦+3= 𝑒 π‘₯ f(π‘₯)= log 6 π‘₯ 𝑦= 𝑒 π‘₯ βˆ’3

9 Graphing Logarithmic Functions
JUST SKETCH GRAPHS

10 Graphing a Logarithmic Function
Example 7: Graph 𝑓 π‘₯ = log 3 π‘₯ Find the inverse function. Create a table of values for 𝑔 π‘₯ Swap the values of x and y to create a table for 𝑓 π‘₯ . 4) Plot the points and connect them with a smooth curve. π‘₯= log 3 𝑦 𝑦= 3 π‘₯ 𝑔(π‘₯)= 3 π‘₯ x -2 -1 1 2 G(x) 1 9 1 3 1 3 9 X 𝟏 πŸ— 𝟏 πŸ‘ 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 =βˆ’5 Rewrite the equation in logarithmic form. 5) 7 2 =49 6) =1 7) 4 βˆ’1 = ) =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 βˆ’5 =32 14 0 =1 log =2 log =0 log =βˆ’1 log = 1 8 = 1 3 =5 β‰ˆ1.079 β‰ˆβˆ’0.288

12 Monitoring Progress 13) 8 log 8 π‘₯ 14) log 7 7 βˆ’3π‘₯
Simplify the expression: 13) 8 log 8 π‘₯ ) log βˆ’3π‘₯ 15) log π‘₯ ) 𝑒 ln 20 17) Find the inverse of 𝑦= 4 π‘₯ 18) Find the inverse of 𝑦= ln (π‘₯βˆ’5) =π‘₯ =βˆ’3π‘₯ =6π‘₯ =20 𝑦= log 4 π‘₯ 𝑦= 𝑒 π‘₯ +5

13 Monitoring Progress Graph the function.
19) 𝑦= log 2 π‘₯ ) 𝑓 π‘₯ = log 5 π‘₯ ) 𝑦= log π‘₯


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