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Aim: How do find the log b a?

Aim: How do find the log b a?. Do Now:. Special Log Values/Properties. Let a and x be positive real numbers such that a  1. because a 0 = 1. log 4 1 = 0. 1. log a 1 = 0. 2. log a a = 1. because a 1 = a. log 4 4 = 1. 3. log a a x = x. because a x = a x.

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Aim: How do find the log b a?

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  1. Aim: How do find the logb a? Do Now:

  2. Special Log Values/Properties Let a and x be positive real numbers such that a 1. because a0 = 1 log4 1 = 0 1. loga 1 = 0 2. logaa = 1 because a1 = a log4 4 = 1 3. loga ax = x because ax = ax log4 43 = 3 Inverse Property 4. because y = ax logax = y inverse  x = ay  substitute logax for y in x = ay

  3. Equivalent Equations 24 = 16 30 = 1 22.585 6 log10 10 = 1 log10 0.1 = -1 log3 1/27 = -3 log16 4096 = 3 log2 1/8 = -3 Converting Logs and Exponents Rewrite the exponential and logarithmic equations logarithmic exponential y = logbx by = x log2 16 = 4 log3 1 = 0 log2 6  2.585 101 = 10 10-1 = 0.1 163= 4096 3-3 = 1/27 2-3= 1/8

  4. 16 = 8x 24 = (23)x 24 = 23x 4 = 3x 4/3 = x Evaluating Logs Find the exponent that makes this statement true Evaluate log8 16 Let x = log8 16 Rewrite log8 16 into exponential form in order to evaluate. Write both sides with base 2 Set exponents equal to each other Solve for x log8 16 = 4/3

  5. 1/49 = 7x 49-1 = 7x (72)-1 = 7x 7-2 = 7x -2 = x Evaluating Logs Evaluate log7 1/49 Let x = log7 1/49 Rewrite log7 1/49 into exponential form in order to evaluate. Write both sides with base 7 Set exponents equal to each other Solve for x log7 1/49 = -2

  6. Find the value of N to the thousandths place in each of the following: log N = 3.9394 log N = -1.7799 If 103.7924 = a, find log a Evaluating Logs (con’t) If log N = 0.6884, what is the value of N? What do I know? • common log - base 10 • exponent is 0.6884 • log N = 0.6884 equivalent to • 100.6884 = N N = 4.879977 . . .

  7. From home screen hit LOG key and enter 79. Close parentheses and hit ENTER . Find log 243 Find log .384 Find log 343 Using Calculator to Find Value of Log10 The logarithmic function with base 10 is called the common log function. If no subscript for base is given assume a base 10 log 100 = 2 Find log 79 = 1.897627091 . . . = 2.385606274 . . . = -.415668 . . . = 4.5944 . . .

  8. characteristic mantissa Find log 120 Finding Common Logarithms Use your calculator to find to the nearest 10,000th. Log 7.83 = Log 78.3 = Log 7830 = Log 783000 = n log 7.83 0.8938 1.8938 3.8938 5.8938 If 1 < a < 10, then 0 < log a < 1 and Log (ax 10n) = log a + n = log 1.2 + log 100 120 = 1.2 x 102 0.0792 + 2 = 2.0792

  9. Natural Logarithmic Function f(x) = logex = ln x, x > 0 because e0 = 1 1. ln 1 = 0 2. ln e = 1 because e1 = e 3. ln ex = x because ex = ex inverse property 5. If ln x = ln y, then x = y The logarithmic function with base e is called the natural log function.

  10. Using Properties of Natural Logarithms Rewrite each expression: = -1 = 2 2. ln e2 3. ln e0 ln ex = x because ex = ex = 0 4. 2ln e = 2 ln e = 1 because e1 = e

  11. 10? = 4 10? = 128 Substitute bx = by  x = y Solve Solving Equations w/logs Alternate method for solving exponential equations Convert each side of equation to power with base 10 Solve 4x = 128 log 4 = 0.60206 log 128 = 2.10721 (100.60206)x = 102.10721 0.60206x = 2.10721 Solve 6x = 280 to nearest thousandth

  12. Use the formula for 106.66 = I x = 4,570,882 times Using Logs to Solve Problems On June 15, 1985, Ted Nugent and the Bad Company played at the Polaris Amphitheater in Columbus, Ohio. Several miles away, the intensity of the music at the concert registered 66.6 decibels. How many times the minimum intensity of sound detectable by the human ear was this sound, if I0 is defined to be 1? Divide by 10, I0 = 1 Rewrite in exponential terms Use the 10xkey Of your calculator

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