Lesson 6.3 Logartithms and Logarithmic Functions(Day 1)

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Presentation transcript:

Lesson 6.3 Logartithms and Logarithmic Functions(Day 1) Learning Goal: (F.LE.A.4) I can explain the relationship between exponents and logarithms. I can evaluate logarithms by using the relationship with exponents. Essential Question: What is the relationship between logarithms and exponents ? Homework Discussion

Developed by John Napier in 1614 “for use in the extensive plane and  spherical trigonometrical calculations necessary for astronomy”. Word Origin: from two ancient Greek terms logos meaning proportion arithmos meaning number Logarithms In this course, we will be using the inverse relationship between  exponential and logarithmic functions to solve problems.

The inverse of an exponential function is a logarithmic function. Example: f(x) = 2x y = 2x (replace f(x) with y) x = 2y (swap x and y) ______ = y (solve for y using definition of logs) f-1(x) = _______ (use inverse notation) Definition of Logarithm: For all positive numbers b, where b ≠ 1, the logby = x if and only if bx = y. (logby ⇒ "the logarithm with base b of y" or "log base b of y") Note: The _____ of the logarithm and the ______ of the  exponent are the same. The logarithm _______ the exponent. Are there any restrictions on the value of y? Justify your answer.

Rewriting Logarithmic Equations logby = x if and only if bx = y Rewrite each equation in exponential form. a. log2 16 = 4 b. log4 1 = 0 c. log12 12 = 1 d. log1/4 4 = −1 Discussion on b and c Logarithm of 1 Logarithm of b with Base b

Rewriting Exponential Functions logby = x if and only if bx = y Rewrite each equation in logarithmic form. a. 52 = 25 b. 10−1 = 0.1 c. 82/3 = 4 d. 6−3 = Your Turn

logby = x if and only if bx = y Evaluating Logarithmic Expressions logby = x if and only if bx = y Evaluate each logarithm. a. log4 64 b. log5 0.2 c. log1/5 125 d. log36 6 Specific Logarithms Common Logarithms: base 10 ; log x = log10x Natural Logarithms: base e ; ln x = logex Example: Evaluate using a calculator. a. log 8 b. ln0.3

Your Turn Rewrite the equation in exponential form. 1. log3 81 = 4 2. log7 7 = 1 3. log14 1 = 0 4. log1/2 32 = −5 Rewrite the equation in logarithmic form. 5. 72 = 49 6. 500 = 1 7. 4−1 = 8. 2561/8 = 2  Evaluate the logarithm. If necessary, use a calculator and round your answer to three decimal places. 9. log2 32 10. log27 3 11. log 12 12. ln 0.75

Practice to Strengthen Understanding Exit Question: What is the relationship between exponential functions and logarithmic functions? Practice to Strengthen Understanding Hmwk #4 BI p314 #5-25,27-32 (show work for #17-24)