4.4 Evaluate Logarithms and Graph Logarithmic Functions Part 2 Definition • Logarithms are the "opposite" of exponentials, • Logs "undo" exponentials. • Logs are the inverses of exponentials. Writing Logarithms _____________________________________________ log b a c -- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - ____________________________________________ You read it: Log base “b” of “a” equals “c” ‘log’ b a c is the operation is the base is the object of the log is what you get when you evaluate the log Exponential Form x b =y Logarithmic Form log x y b = Evaluating logarithms now you try some! • • • • 4 16 Log 4 16 = 2 x 0 Log 5 1 = 5 1 x Log 16 4 = ½ (because 16 16 4 1/2 = 4) x undefined 3 1 Log 3 (-1) = x (Think of the graph of y = 3x) You should learn the following general forms!!! • Log a 1 = 0 because a0 = 1 • Log a a = 1 because 1 a =a • Log a ax = x because ax = ax Common logarithms •log x = log 10 x • Understood base 10 if nothing is there. Common Logs and Natural Logs with a calculator log10 button lne button Finding Inverses • Find the inverse of: • y = log3x • By definition of logarithm, the inverse is y=3x • OR write it in exponential form and switch the x & y! y 3 =x x 3 =y Example 1: • Write 53 = 125 in logarithmic form. • Write log381 = 4 in exponential form. Try This: Complete the table. #1 Exponential Form Logarithmic Form #2 25 = 32 #3 #4 3-2 = 1/9 log101000 = 3 Log164 = 1/2 Lets look at their graphs y 10 x y log10 x log10 y x y=x To Evaluate Logs without a Calculator • Change the log to an exponential. 1. log2 32 = x 2. log4 2 = x Solve for x. Change the log to an exponential. 1. log2 64 = x 2. logx 343 = 3 Evaluate without a calculator: Change the log to an exponential. 1. log 2 8 = x 2. log 2 1 = x 3. Find the value of k : k = log 4. Find the value of k : ½ = log k 9 5. Find the value of k : 3 = log 7 k 9 3 Common Logarithms 10 are called • Logarithms with base ______ common logarithms. • Sometimes the base is assumed and not written. • Thus, if you see a log written without a base, 10 you assume the base is _______. • The log button the calculator uses base 10 _____. Use your calculator to evaluate: 1. log 51 1.71 2. log 4 0.6 3. log 0.215 – 0.67 Which means 101.71 51 Do You Know What X is? Change the exponential to a log. Then use calculator. 4. Solve for x: 10x = 728 5. Solve for x: 1 10 1085 x Remember e? e 2.718 Natural Logarithm • A natural logarithm is a logarithm with base e, denoted by ln. • A natural logarithm is the inverse of an exponential function with base e. log e x ln x Exponential Form e 7.389 2 Logarithmic Form ln 7.389 2 Lets look at their graphs ye y ln x x ln y x y=x Write as exponent or log. y ln x 1. e 4 x 2. ln 56.3 4.03 Evaluate f(x)=ln x to the nearest thousandth for each value of x below: 3. x 2 0.693 1 4. x 2 – 0.693 5. x 1 ? (see graph) 13. Find the inverse of y = ln(x+1) y = ex - 1 14. Find the inverse of y = 5x . y = log5x Homework Book Pg. 147 16 - 24 all Pg. 148 13 – 21 all