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5.3.1 5.3: Logarithms Looking at the graph of f ( x) = b x , we can see it is one-to-one. So every exponential function has an inverse. Example 1: Consider the function f ( x) = 3x . Then f (1) = 3 , f (2) = 9 , f (3) = 27 , and f (4) = 81 . This function has an inverse. So f −1 (3) = 1 , f −1 (9) = 2 , f −1 (27) = 3 , and f −1 (81) = 4 . We call this inverse function a logarithmic function and denote it f −1 ( x) = log 3 x . So f −1 (3) = log 3 3 = 1 and f −1 (9) = log 3 9 = 2 . Also log 3 27 = 3 and log 3 81 = 4 . Every exponential function has an inverse. The inverses of exponential functions are called logarithmic functions (logarithms or logs for short). Definition: log b x = y means b y = x . The functions f ( x) = b x and g ( x) = log b x are inverses of each other. b is called the base of the logarithm. 5.3.2 Evaluating logarithms: Example 2: Evaluate log 2 8 . In other words, log 2 8 is a number. What number is it? This question is asking us to find a certain exponent. Specifically, “what exponent must I put on the 2 to give me 8?” Said another way, “2 raised to what power is 8?” Examples: log 2 32 = _______ log 5 ( 5 ) = _______ 4 log 5 1 = _______ ⎛1⎞ log 3 ⎜ ⎟ = _______ ⎝9⎠ log10 100 = _______ ⎛ 1 ⎞ log10 ⎜ ⎟ = _______ ⎝ 1000 ⎠ log 3 3 = _______ log 1 1 = _______ 2 log 3 3 = _______ log 2 ( 16 ) = _______ 3 ⎛ 1 ⎞ log 3 ⎜ ⎟ = _______ ⎝ 3⎠ log 64 8 = _______ 5.3.3 Evaluating more complicated logarithms: Example 3: log 4 32 = _______ Example 4: ⎛ 1 ⎞ log 9 ⎜ ⎟ = _______ ⎝ 27 ⎠ The natural logarithm: Remember we said e was a very important number? It is so important that the logarithmic function of base e has its own special notation and its own button on your calculator. The logarithm of base e is called the natural logarithm, which is abbreviated “ln”. log e x = ln x Example 5: ln e 4 = _______ Example 6: ⎛1⎞ ln ⎜ 3 ⎟ = _______ ⎝e ⎠ 5.3.4 Example 7: Evaluate ln e . Example 8: ⎛ 1 ⎞ Simplify ln ⎜ ⎟. 3 5 ⎝ e ⎠ Example 9: Evaluate ln1 . Example 10: Evaluate log 2 (−4) . Example 11: Evaluate log 5 0 . IMPORTANT: You cannot apply a logarithm to zero or to a negative number!!! Exponential and logarithmic forms for an equation: Remember, log b x = y means b y = x . Logarithmic form: log b x = y Exponential form: b y = x 5.3.5 Example 12: Convert each of the following to exponential form. a) log10 1000 = 3 b) log a 178 = w c) log 7 ( y − 3) = x d) log x 6 = y 2 + 2 Example 13: Convert each of the following to logarithmic form. a) 7 x = 23 b) y x −1 = 8 c) x8 = u d) ( x − 3) 2 = 6 about the relationship between f ( x) = b x and g ( x) = log b x : Because f ( x) = b x and g ( x) = log b x are inverses of one another, f ( g ( x)) = x and g ( f ( x)) = x . This gives us… log b b x = x b logb x = x 5.3.6 Example 14: Simplify log 3 3 x+ 5 . Example 15: Simplify log 2 212 . Example 16: Simplify ln e−2 . Example 17: Simplify 5log5 y . Example 18: Simplify 3log3 Example 19: Simplify eln( x 2 2 . +1) . The common logarithm: Often log is used to mean log10 . The logarithm of base 10 is called the common logarithm. Example 20: Evaluate log 3 10 . Solving simple logarithmic equations: Example 21: Solve for x. log 2 ( x − 1) = 5 Example 22: Solve for x. log x 7 = 1 2 5.3.7 Properties of Logarithms Laws (properties, rules) of logarithms: 1. log b b = 1 2. log b 1 = 0 3. log b ( PQ) = log b P + log b Q ⎛P⎞ 4. log b ⎜ ⎟ = log b P − log b Q ⎝Q⎠ ⎛1⎞ Note: this results in log b ⎜ ⎟ = − log b Q . ⎝Q⎠ 5. log b P n = n log b P 6. blogb P = P 7. log b b P = P log b x (Change of base formula) log b a ln x Note: this results in log a x = . ln a 8. log a x = Example 1: Simplify log 2 56 − log 2 7 . Example 2: Simplify log 6 4 + log 6 9 . Example 3: Simplify log 50 − log 2 + log 4 . 5.3.8 10 Example 4: Simplify log 2 8 . Example 5: Simplify 5 2 log 5 3 . Example 6: Simplify e6ln 2 . ⎛3x⎞ Example 7: Simplify log 6 ⎜⎜ ⎟⎟ . 36 ⎝ ⎠ Errors to avoid: log a ( x + y ) ≠ log a x + log a y log a x ≠ log a x − log a y log a y (log a x)3 ≠ 3log a x Putting the properties together: Example 8: Use the properties of logarithms to expand the expression as much as possible. log 3 ( x( x + 4)) Example 9: Use the properties of logarithms to expand the expression as much as possible. ⎛ x+2⎞ ln ⎜ ⎟ ⎝ e ⎠ 5.3.9 Example 10: Use the properties of logarithms to expand the expression as much as possible. ⎛ ab3 ⎞ ln ⎜ 2 ⎟ ⎝c d ⎠ Example 11: Use the properties of logarithms to expand the expression as much as possible. ⎡ x+5 ⎤ ln ⎢ 2 ⎣ x − 4 ⎥⎦ Example 12: Use the properties of logarithms to expand the expression as much as possible. ⎡ ⎤ x +1 log10 ⎢ 2 ⎥ ⎣ x ( x − 3) x + 7 ⎦ Example 13: Use the properties of logarithms to expand the expression as much as possible. ln ( x − 3) 2 ( x + 9) 4 x 6 ( x − 10) 5.3.10 Example 14: Rewrite as a single logarithm with a coefficient of 1. log 3 x + log 3 2 Example 15: Rewrite as a single logarithm with a coefficient of 1. log a b − c log a d + r log a s Rewrite as a single logarithm with a coefficient of 1. 2 ln x − 5ln( x + 1) − 12 ln( x − 3) + ln( x + 2) Rewrite as a single logarithm with a coefficient of 1. 1 [3log 5 ( x + 2) − 2 log 5 ( x − 1) + log 5 x − 4 log 5 ( x − 7)] 2 Example 16: Rewrite ln 60 so that it contains only logs that are being applied to prime numbers. Example 17: Rewrite log 72 so that it contains only logs that are being applied to prime numbers. 5.3.11 Evaluating logs on your calculator: Example 23: Evaluate log10 72 on your calculator. Example 24: Evaluate ln12 on your calculator. Example 18: Use a calculator to approximate log 2 17 to the nearest hundredth. Example 19: Use a calculator to evaluate log 7 12 to the nearest thousandth. About calculators and books: • Most calculators and some books use o ln for natural log (base e). o log for log10 . • Some math books (usually very advanced ones or those from other countries) o use log for natural log. o That’s all…logs of other bases are useless. • Some books (this is my preference) o use ln for natural log. o use log10 for log10 . o never write log by itself—too confusing!!