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## Exponential and Logarithmic Functions

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**ExponentialandLogarithmic Functions**By: Jeffrey Bivin Lake Zurich High School jeff.bivin@lz95.org Last Updated: January 30, 2008**With your Graphing Calculatorgraph each of the following**y = 2x y = 3x y = 5x y = 1x Determine what is happening when the base is changing in each of these graphs. Jeff Bivin -- LZHS**y = 3x**y = 2x Jeff Bivin -- LZHS**y = 3x**y = 5x y = 2x y = 1x Jeff Bivin -- LZHS**y = 3x**y = 5x y = 2x y = 4x y = 10x y = (3/2)x Determine where each of the following would lie? y=10x y=4x y = (3/2)x y = 1x Jeff Bivin -- LZHS**Exponential**graphs with translations Jeff Bivin -- LZHS**f(x) = 2x**Jeff Bivin -- LZHS**f(x) = 2x-3**x - 3 = 0 x = 3 (3, 1) 3 Jeff Bivin -- LZHS**f(x) = 2x+2 - 3**x + 2 = 0 x = -2 3 2 (-2, -2) y = -3 Jeff Bivin -- LZHS**flip**flip f(x) = -(2)x-4 – 2 x - 4 = 0 x = 4 4 2 y = -2 (4, -3) Jeff Bivin -- LZHS**Compound Interest**You deposit $5000 into an account that pays 4.5 % interest. What is the balance of the account after 10 years if the interest is compounded quarterly? A = Final amount = unknown P = Principal = $5000 r = rate of interest = .045 n = number of times compounded per year = 4 t = number of years compounded = 10**Compound Interest**You deposit $5000 into an account that pays 4.5 % interest. What is the balance of the account after 10 years if the interest is compounded quarterly? A = unknown P = $5000 r = .045 n = 4 t = 10**Compound Interest**You deposit $5000 into an account that pays 4.5 % interest. What is the balance of the account after 10 years if the interest is compounded quarterly? weekly? A = unknown P = $5000 r = .045 52 n = 4 t = 10**Exponential**DECAY**With your Graphing Calculatorgraph each of the following**y = (1/2)x y = (1/3)x y = 1x Determine what is happening when the base is changing in each of these graphs. Jeff Bivin -- LZHS**y = (1/3)x**y = 5x y = 2x y = 3x y = (½)x y = 1x Jeff Bivin -- LZHS**f(x) = 2-x = (1/2)x**(0, 1) Jeff Bivin -- LZHS**f(x) = (½)x-3 - 2 = (2)-x+3 - 2**x - 3 = 0 x = 3 3 2 (3, -1) y = -2 Jeff Bivin -- LZHS**A new Number**We could use a spreadsheet to determine an approximation.**y = 3x**y = 2x Graph y = ex y = ex Jeff Bivin -- LZHS**Graph:**y = ex+2 y = ex+2 y = ex x + 2 = 0 x = -2 Jeff Bivin -- LZHS**Compound Interest-continuously**You deposit $5000 into an account that pays 4.5 % interest. What is the balance of the account after 10 years if the interest is compounded continuously? A = Final amount = unknown P = Principal = $5000 r = rate of interest = .045 t = number of years compounded = 10**Compound Interest-continuously**You deposit $5000 into an account that pays 4.5 % interest. What is the balance of the account after 10 years if the interest is compounded continuously? A = unknown P = $5000 r = .045 t = 10**Bacteria Growth**You have 150 bacteria in a dish. It the constant of growth is 1.567 when t is measured in hours. How many bacteria will you have in 7 hours? y = Final amount = unknown n = initial amount = 150 k = constant of growth = 1.567 t = time = 7**Bacteria Growth**You have 150 bacteria in a dish. It the constant of growth is 1.567 when t is measured in hours. How many bacteria will you have in 7 hours? y = unknown n = 150 k = 1.567 t = 7**y = 2x**x = 2y INVERSE Jeff Bivin -- LZHS**y = 2x**x = 2y INVERSE How do we solve this exponential equation for the variable y? ? Jeff Bivin -- LZHS**LOGARITHMS**exponential logarithmic b > 0 A > 0 Jeff Bivin -- LZHS**exponential**logarithmic Jeff Bivin -- LZHS**Evaluate**Jeff Bivin -- LZHS**Evaluate**Jeff Bivin -- LZHS**Evaluate**Jeff Bivin -- LZHS**Evaluate**Jeff Bivin -- LZHS**Evaluate**Jeff Bivin -- LZHS**Evaluate**Jeff Bivin -- LZHS**y = 2x**x = 2y INVERSE y=log2x Jeff Bivin -- LZHS**x = 2y**y = log2x y = log3x y = log5x Jeff Bivin -- LZHS**x = (½)y**y = log½x Jeff Bivin -- LZHS**Solve for x**log2(x+5) = 4 24= x + 5 16= x + 5 11= x Jeff Bivin -- LZHS**Solve for x**logx(32) = 5 x5= 32 x5 = 25 x = 2 Jeff Bivin -- LZHS**Evaluate**log3(25) = u 3u= 25 3u = 52 ?????? Jeff Bivin -- LZHS**Change of Base Formula**if b = 10 Jeff Bivin -- LZHS**Evaluate**log3(25) = 2.930 Jeff Bivin -- LZHS**Evaluate**log5(568) = 3.941 Jeff Bivin -- LZHS**Properties of Logarithms**• Product Property • Quotient Property • Power Property • Property of Equality Jeff Bivin -- LZHS**Product Property**multiplication addition multiplication addition Jeff Bivin -- LZHS