1. Aim: Evaluating Logs Course: Alg. 2 & Trig. Aim: How do find the log b a? Do Now:

2. Aim: Evaluating Logs Course: Alg. 2 & Trig. Special Log Values/Properties Let a and x be positive real numbers such that a  1. 1. log a 1 = 0 2. log a a = 1 3. log a a x = x because a 0 = 1 because a 1 = a because a x = a x log 4 4 3 = 3 log 4 4 = 1 log 4 1 = 0 4. Inverse Property log a x = y because y = a x x = a y  inverse  substitute log a x for y in x = a y

3. Aim: Evaluating Logs Course: Alg. 2 & Trig. Converting Logs and Exponents log 2 16 = 4 Rewrite the exponential and logarithmic equations log 3 1 = 0 log 2 6  2.585 2 4 = 16 10 1 = 10 10 -1 = 0.1 16 3 = 4096 logarithmic exponential y = log b x b y = x Equivalent Equations 3 0 = 1 2 2.585  6 log 10 10 = 1log 10 0.1 = -1 log 16 4096 = 3 3 -3 = 1/27 2 -3 = 1/8 log 3 1/27 = -3 log 2 1/8 = -3

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

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

6. Aim: Evaluating Logs Course: Alg. 2 & Trig. Evaluating Logs (con’t) If log N = 0.6884, what is the value of N? exponent is 0.6884 common log - base 10 What do I know? log N = 0.6884 equivalent to 10 0.6884 = N N = 4.879977... Find the value of N to the thousandths place in each of the following: log N = 3.9394log N = -1.7799 If 10 3.7924 = a, find log a

7. Aim: Evaluating Logs Course: Alg. 2 & Trig. Using Calculator to Find Value of Log 10 Find log 79 From home screen hit LOG key and enter 79. Close parentheses and hit ENTER. = 1.897627091... = 2.385606274... = -.415668... Find log 243 Find log.384 Find log 34 3 = 4.5944... 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

8. Aim: Evaluating Logs Course: Alg. 2 & Trig. Finding Common Logarithms Use your calculator to find to the nearest 10,000th. Log 7.83 = Log 78.3 = Log 7830 = Log 783000 = 0.8938 1.8938 3.8938 5.8938 n If 1 < a < 10, then 0 < log a < 1 and Log (a x 10 n ) = log a + n Find log 120 120 = 1.2 x 10 2 = log 1.2 + log 100 0.0792+ 2= 2.0792 log 7.83 characteristicmantissa

9. Aim: Evaluating Logs Course: Alg. 2 & Trig. Natural Logarithmic Function f(x) = log e x = ln x, x > 0 1. ln 1 = 0 2. ln e = 1 3. ln e x = x because e 0 = 1 because e 1 = e because e x = e x The logarithmic function with base e is called the natural log function. inverse property 5. If ln x = ln y, then x = y

10. Aim: Evaluating Logs Course: Alg. 2 & Trig. Using Properties of Natural Logarithms 2. ln e 2 3. ln e 0 Rewrite each expression: ln e x = x because e x = e x = -1 = 2 = 0 4. 2ln e= 2 ln e = 1 because e 1 = e

11. Aim: Evaluating Logs Course: Alg. 2 & Trig. Solve b x = b y  x = y Substitute Solving Equations w/logs Solve 4 x = 128 Convert each side of equation to power with base 10 10 ? = 4 10 ? = 128 Alternate method for solving exponential equations log 4 = 0.60206 log 128 = 2.10721 (10 0.60206 ) x =10 2.10721 0.60206x = 2.10721 Solve 6 x = 280 to nearest thousandth

12. Aim: Evaluating Logs Course: Alg. 2 & Trig. 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 I 0 is defined to be 1? Use the formula for Divide by 10, I 0 = 1 10 6.66 = I Rewrite in exponential terms x = 4,570,882 times Use the 10 x key Of your calculator