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Unit 5: Modeling with Exponential & Logarithmic Functions. Ms. C. Taylor. Warm-Up. Identify the value of b in the following:. Graphing Exponential Equations. The graph will approach the axis but will never touch. Asymptote for the function will approach the x-axis.
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Unit 5: Modeling with Exponential & Logarithmic Functions Ms. C. Taylor
Warm-Up • Identify the value of b in the following:
Graphing Exponential Equations • The graph will approach the axis but will never touch. • Asymptote for the function will approach the x-axis. • Asymptote for the inverse function will approach the y-axis.
Warm-Up • Rewrite using exponent rules
Logarithms • Suppose b>0 and b≠1. For x>0, there is a number y such that if and only if
LogarithmicExponential Inequality • If • If
Property of Equality for Logarithmic Functions • If b is a positive number other than 1, then if and only if • Example: If , then
Property of Inequality for Logarithmic Functions • If , then if and only if, and if and only if • If , then
Product Property of Logarithms • For all positive numbers m, n, and b, where b≠1,
Example #1 • Expand the following logarithms:
Example #2 • Use to approximate the value of • Use to approximate the value of
Quotient Property of Logarithms • For all positive numbers m, n, and b, where ,
Example #3 • Expand the following logarithms:
Example #4 • Use and to approximate • Use and to approximate
Power Property of Logarithms • For any real number p and positive numbers m and b, where ,
Examples • Given , approximate the value of • Given , approximate the value of
Warm-Up • Expand the following:
Change of Base Formula • For all positive numbers, a, b, and n, where and ,
Examples • Express in terms of common logarithms. Then approximate its value to four decimal places. • Express in terms of common logarithms. Then approximate its value to four decimal places.
Equivalent Expressions • If something has an e in it then that will become a ln. • If something has an ln in it then it will become e raised to a power.
Warm-Up • Evaluate the following
Warm-Up • Use the properties of logarithms to rewrite: