Exponential and Log Functions BY: Brandon, Ashley, Alicia

Vocab Exponential equation: An equation containing one or more expressions that have a variable as an exponent. Logarithmic equation: An equation with a logarithmic expression that contains a variable. These equations can be used by applying properties of logarithms we are familiar with. Asymptote: A line that a graphed function approaches as the value of x gets very large, or very small.

Properties of Logarithms 1.Log b 1 =0 2.Log b b =1 3.Log b b^x =x 4.b^ log b x =x Angle Method: 5.If b^x = b^y, then x=y log b x = y b^y = x

Properties of Logarithms Cont. Log. of a Product: Log of a Quotient: Log b (xy) + log b x +log b y Log b (x/y) = log b x - log b y Ex. log(42) = log 6 * 7 = log6 +log7 Ex. log(15) = log30 - log15 Ex. Log 3 y/8 = log 3 y - log 3 8 Log. of a power: Change of Base Formula: Log b (x^n) = n log b x Log a b = log b / log a Ex. log b 4√x = log a x^⅓ = ⅓ log a x Ex. log 9 42 = log 42 / log 9

Exponential: Growth and Decay Equations Formula: f(x)= b^x Percent Increase & Decrease: A(t)= a(1+r)^t Compound Interest Formula: A= P (1+ r/n)^n*t 1+r= growth1-r= decay Ex. y= 2^x /41/2124

Graphing Exponential and Log Functions Table Method To graph an exponential function you would make a table and plug in values. Since an exponential function is the inverse of a log Function, a log function on a graph is the inverse of an Exponential function on a graph, therefor to graph a Log function you would make a table using the exponential Function and just switch X and Y.

Graphing Exponential and Log Functions Transformation Method When using transformations you will have to alter the points of a basic exponential function f(x)=2^x by transforming the known points of this function. Formula: y = a(base)^(x-h)+v In the formula stated above “a” is the vertical stretch or compression, “h” is the horizontal shift and “v” is the vertical shift Example: 2^x to 2(2^(x+3))+4 This transformation goes left 3, up 4 and has a vertical stretch by a factor of 2

Transforming basic function points Example: *Transforming basic exponential function points with a vertical stretch of 2 Xf(x)= 2^x -21/4 1/ Xf(x)=2(2 ^x) -21/

Solving Equations 1.log 4 -4p= log 4 (3p+7)3. Log 4 16 ^(x-3) 2.log 9 (3k-9)= log 9 (2k+9) Graph the function then its inverse4. log 3 7 = y=3^x. 6. Log 4 x log 3 10 = 2.1 log 3 8 = 1.9 Find log 3 64

Solving Equations (cont) Find inverse 7. 2^4=16 8. log = ^2 = log 5 625= 4 Solve 11. log 10 x = 1,000, log45x- log3= 1

Answers 1.p=1 2.k=18 3.2x y=log 3 x 6.f(x)= 4 x 7. log 2 16= = log = = x=6 12. x= 2/3

Exponential Growth and Decay Word Problems: 1.Sharquisha owns a spa. Her first year she made $10,500 and each of the following years her profit increased 9%. 1.Marsha deposited $1,600 into a bank account. Find the balance after 3 years if the account pays 2.5% annual interest compounded monthly.

Answer: Word Problems 1.A= 10,500(1.09)^t 1.A= $ A= 1600(1+.025/12)^12(3)