11.4 Logarithmic Functions Objectives: Evaluate expressions involving logarithms. Solve equations and inequalities with logs. Graph logarithmic functions and inequalities.

Exponential Functions and Logarithmic Functions are Inverses. 2^x=y y -1 2^-1=y 2 2^2=y 6 2^6=y y 2^y=x x 2^y=½ ½ 2^y=4 4 2^y=64 64 The exponent y is called the logarithm, base 2 of x, written as: log2x = y It is read, “the log base 2 of x is = to y”

A log is an EXPONENT!!!! Exponential Equation Logarithmic Equation x = a^y y = logax x→number a → base y → exponent Definition of The logarithm function y = logax, where Logarithm: a>0 and a≠1, is the inverse of the exponential function y=a^x. SO, y=logax if and only if x = a^y.

Ex. 1) Exponential Equation Log Equation a.) 3²=9 a.) log39=2 b.) 10^4=10,000 b.) log1010,000=4 c.) 4^0=1 c.) log41=0 d.) 3^-3=1/27 d.) log31/27=-3 e.) 25^½=5 e.) log255=½ Ex. 2) Evaluate: a.) log71/49=y b.) log⅓27=y c.) log6x=2

Composites * logaa^x * a^logax=x Properties of Logs: Property Definition Product logbmn = logbm + logbn Quotient logbm/n = logbm – logbn Power logbm^p = p · logbm Equality If logbm = logbn, then m=n

Ex. 3) Evaluate a.) log55^8 b.) 5^(log5(2x+2)) Ex. 4) Evaluate a.) log7(2x+1) = log7(3x-5) b.) log8(x²-14) = log8(5x) Ex. 5) Evaluate a.) logb15,625^1/6) = -⅓ b.) log3(4x+5) – log3(3-2x) = 2

Ex. 6) Graph y = log2(x-1) Ex. 7) Graph y < log3(x+1)