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5.3 Intro to Logarithms 2/27/2013

5.3 Intro to Logarithms 2/27/2013. Definition of a Logarithmic Function. For y > 0 and b > 0, b ≠ 1, log b y = x if and only if b x = y Note: Logarithmic functions are the inverse of exponential functions Example: log 2 8 = 3 since 2 3 = 8 Read as: “log base 2 of 8”.

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5.3 Intro to Logarithms 2/27/2013

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  1. 5.3 Intro to Logarithms2/27/2013

  2. Definition of a Logarithmic Function Fory> 0 and b > 0, b ≠ 1, logby = x if and only if bx=y Note: Logarithmic functions are the inverse of exponential functions Example: log28 = 3 since 23= 8 Read as: “log base 2 of 8”

  3. Exponent Exponent Base Base Location of Base and Exponent in Exponential and Logarithmic Forms Logarithmic form:x= logby Exponential Form: bx=y

  4. Basic Logarithmic Properties Involving One logby = x if and only if bx = y • Logbb = __ because 1 is the exponent to which b must be raised to obtain b. (b1 = b). • Logb 1 = __ because 0 is the exponent to which b must be raised to obtain 1. (b0 = 1). 1 0

  5. Base 10 log 10x = log x is called a common logarithm Base “e” log e x = ln x is called the natural logarithm or “natural log” Popular Bases have special names

  6. e and Natural Logarithm e is the natural base and is also called “Euler’s number” : an irrational number (like ) and is approximately equal to 2.718281828... Real Life Use: Compounding Interest problem Remember the formula as n approaches + The Natural logarithmof a number x(written as “ln (x)”) is the power to which e would have to be raised to equal x. For example, ln(7.389...) is 2, because e2=7.389 Note: and ln(x) are inverse functions.

  7. Inverse properties Since are inverse functions. and Proof: Then Proof: Since and ln(x) are inverse functions. and

  8. Example 1 24 16 = b. log7 1 0 70 1 = = c. log5 5 1 51 5 = = d. log0.01 2 0.01 = = – 10 2 – e. log1/4 4 1 4 = = – 1 1 – 4 Rewrite in Exponential Form logby = x is bx= y EXPONENTIAL FORM LOGARITHMIC FORM a. log2 16 4 =

  9. Example 1 Rewrite in Exponential Form EXPONENTIAL FORM LOGARITHMIC FORM log e x = ln x = f. ln = g. ln

  10. Example 2 Rewrite in Logarithmic Form Form logby = x is bx= y EXPONENTIAL FORM LOGARITHMIC FORM a. = b. = c. = d. =

  11. Example 3 1 41/2 2 = Guess, check, and revise. log4 2 = 2 4? 64 = What power of 4 gives 64? 43 64 = Guess, check, and revise. log4 64 3 = 4? 2 = What power of 4 gives 2? Evaluate Logarithmic Expressions logby = x is bx= y Evaluate the expression. a. log4 64 b. log4 2

  12. Example 3 ? 1 9 = What power of gives 9? 3 1 3 2 1 – 9 = Guess, check, and revise. 3 log1/3 9 2 = – Evaluate Logarithmic Expressions c. log1/3 9 d. Since

  13. Example 4 Simplifying Exponential Functions a. Since = 5 b. Since =

  14. Example 4 Simplifying Exponential Functions c. Since = 6 d. Since =

  15. Homework WS 5.3 odd problems only

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