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ch.5 Logarithmic Functions

ch.5 Logarithmic Functions - Skills Refresher Ch 5 pg 219...

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Unformatted text preview: Skills Refresher Ch. 5 pg. 219 Logarithms Chapter 5 Logarithmic Functions 5.1 LOGARITHMS AND THEIR PROPERTIES Key Points • Using logarithms to solve exponential equations • The definition of the logarithm function • The equivalence of exponential and logarithmic expressions • The inverse relationship between the y = log(x) and y = 10x What is a Logarithm? If x is a positive number, log x is the exponent of 10 that gives x . In other words, if y = log x then 10 y = x . If x is a positive number, log x is the exponent of 10 that gives x . In other words, if y = log x then 10 y = x . Functions Modeling Change: A Preparation for Calculus, 4th Evaluate each of the following. Evaluate each of the following. • log10 log10 • log 100 log 100 • log 1000 log 1000 • log 10000 log 10000 • log (1/10) log (1/10) • log (1/100) log (1/100) • log (1/1000) log (1/1000) • log 1 log 1 • 1 1 because 10 because 10 1 = 10 = 10 • 2 2 because 10 because 10 2 = 100 = 100 • 3 3 because 10 because 10 3 = 1000 = 1000 • 4 4 because 10 because 10 4 = 10000 = 10000 • – – 1 1 because 10 because 10-1-1 = 1/10 = 1/10 • – – 2 2 because 10 because 10-2-2 = 1/100 = 1/100 • – – 3 3 because 10 because 10-3-3 = 1/1000 = 1/1000 • because 10 because 10 = 1 = 1 Graph of Graph of f f ( ( x x ) = log ) = log x x x f ( x ) .01-2 .1-1 1 10 1 100 2 Note that the graph of Note that the graph of y y = log = log x x is the is the graph of y = 10 graph of y = 10 x x reflected through the reflected through the line line y y = = x x . This suggests that these are . This suggests that these are inverse functions. inverse functions. Solving Exponential Equations Solving Exponential Equations Using The Inverse Property Using The Inverse Property log( log(10 x ) = x • Solve Solve the equation 10 the equation 10 x = 35 = 35 • Take the common log of both sides Take the common log of both sides log log 10 10 x = log 35 = log 35 • Using the inverse property log( Using the inverse property log(10 x ) = x this simplifies to x = log 35 • Using the calculator to estimate log 35 we have x ≈ 1.54 Solving Logarithmic Equations Solving Logarithmic Equations Using The Inverse Property Using The Inverse Property 10 10 log log x x = = x x • Solve Solve...
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