Exponential and Logarithmic Functions By: Jeffrey Bivin Lake Zurich High School Last Updated: January 30, 2008
With your Graphing Calculator graph each of the following y = 2 x y = 3 x y = 5 x y = 1 x Determine what is happening when the base is changing in each of these graphs. Jeff Bivin -- LZHS
y = 2 x Jeff Bivin -- LZHS xy = 2 x y = 3 x -21/41/9 ½1/ y = 3 x
y = 2 x Jeff Bivin -- LZHS xy = 5 x y = 1 x -21/251 1/ y = 3 x y = 5 x y = 1 x
y = 2 x Jeff Bivin -- LZHS y = 3 x y = 5 x y = 1 x Determine where each of the following would lie? y=10 x y=4 x y = (3/2) x y = 10 x y = 4 x y = (3/2) x
Jeff Bivin -- LZHS
f(x) = 2 x Jeff Bivin -- LZHS
f(x) = 2 x-3 Jeff Bivin -- LZHS
f(x) = 2 x Jeff Bivin -- LZHS
f(x) = -(2) x-4 – 2 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? A = unknown P = $5000 r =.045 n = 4 t = 10 weekly? 52
With your Graphing Calculator graph each of the following y = 1 x y = (1/2) x y = (1/3) x Determine what is happening when the base is changing in each of these graphs. Jeff Bivin -- LZHS
y = 2 x Jeff Bivin -- LZHS x y = (½) x y = (1/3) x ½1/3 2¼1/9 31/81/27 y = 3 x y = 5 x y = 1 x y = (1/3) x y = (½) x
f(x) = 2 -x = (1/2) x Jeff Bivin -- LZHS
f(x) = (½) x = (2) -x Jeff Bivin -- LZHS
A new Number We could use a spreadsheet to determine an approximation.spreadsheet
A new Number
y = 2 x Jeff Bivin -- LZHS xy = 2 x y = 3 x -2¼1/9 ½1/ y = 3 x y = e x Graph y = e x
Jeff Bivin -- LZHS y = e x y = e x+2 Graph: y = e x+2 x + 2 = 0 x = -2
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 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 = t = time = 7
Bacteria Growth You have 150 bacteria in a dish. It the constant of growth is when t is measured in hours. How many bacteria will you have in 7 hours? y = unknown n = 150 k = t = 7
y = 2 x Jeff Bivin -- LZHS xy -21/4 ½ xy 1/4-2 ½ x = 2 y
How do we solve this exponential equation for the variable y? y = 2 x x = 2 y Jeff Bivin -- LZHS
LOGARITHMS exponentiallogarithmic b > 0 A > 0 Jeff Bivin -- LZHS
exponentiallogarithmic 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 = 2 x Jeff Bivin -- LZHS xy -21/4 ½ xy 1/4-2 ½ x = 2 y y=log 2 x
Jeff Bivin -- LZHS x y = log 2 x 1/4-2 ½ y = log 2 x y = log 3 x y = log 5 x x = 2 y
Jeff Bivin -- LZHS x y = log ½ x 1/42 ½ y = log ½ x x = (½) y
Solve for x log 2 (x+5) = = x = x = x Jeff Bivin -- LZHS
Solve for x log x (32) = 5 x 5 = 32 x 5 = 2 5 x = 2 Jeff Bivin -- LZHS
Evaluate log 3 (25) = u 3 u = 25 3 u = 5 2 ?????? Jeff Bivin -- LZHS
Change of Base Formula Jeff Bivin -- LZHS
Evaluate log 3 (25) = Jeff Bivin -- LZHS
Evaluate log 5 (568) = Jeff Bivin -- LZHS
Properties of Logarithms Product Property Quotient Property Power Property Property of Equality Jeff Bivin -- LZHS
Product Property multiplicationaddition multiplicationaddition Jeff Bivin -- LZHS
Product Property Jeff Bivin -- LZHS
Quotient Property divisionsubtraction divisionsubtraction Jeff Bivin -- LZHS
Quotient Property Jeff Bivin -- LZHS
Power Property log b (m p ) log b (m p ) = plog b (m) p Jeff Bivin -- LZHS
Power Property Jeff Bivin -- LZHS
Property of Equality Jeff Bivin -- LZHS
Expand product property power property Jeff Bivin -- LZHS
Expand quotient property product property power property Jeff Bivin -- LZHS
Expand quotient property product property power property distributive property Jeff Bivin -- LZHS
Condense power property product property quotient property Jeff Bivin -- LZHS
Condense group / factor product property quotient property Power property Jeff Bivin -- LZHS
Condense re-organize group product property Power property quotient property Jeff Bivin -- LZHS
Solve for x Jeff Bivin -- LZHS Property of Equality
Solve for x checks! Jeff Bivin -- LZHS
Solve for x Jeff Bivin -- LZHS
Solve for n Jeff Bivin -- LZHS Condense left side Property of Equality
Solve for n checks! Jeff Bivin -- LZHS
Solve for n Jeff Bivin -- LZHS
Solve for x Jeff Bivin -- LZHS Condense left side Convert to exponential form
Solve for x checks! fails The argument must be positive Jeff Bivin -- LZHS
Solve for x Jeff Bivin -- LZHS
Solve for x Jeff Bivin -- LZHS Convert to exponential form
Solve for x checks! Jeff Bivin -- LZHS
Solve for x Jeff Bivin -- LZHS
Solve for x Jeff Bivin -- LZHS
Solve for x Jeff Bivin -- LZHS
Solve for x Jeff Bivin -- LZHS
Solve for x Jeff Bivin -- LZHS
Solve for x Jeff Bivin -- LZHS
Solve for x Jeff Bivin -- LZHS
Solve for x Jeff Bivin -- LZHS
Solve for x Jeff Bivin -- LZHS