5.4: Logarithmic Functions and Models December 8, 2008 5.4: Logarithmic Functions and Models

5.4 Objectives Evaluate the common logarithmic function Solve basic exponential and logarithmic equations Evaluate logarithms with different bases Solve general exponential and logarithmic equations

Common logarithm log x = k if x = 10k Properties of this include: log 10x = x 10logx = x

Base a logarithm logax = k if and only if.. x=ak a>0 and not equal to 1 k is a real number

Properties of base a logarithms logaax=x (for any real number x) alogax=x (for any positive number x)

Connecting to Inverses The inverse function of f(x)=ax Is….. loga

HOW TO SOLVE LOGARITHMIC EQUATIONS In the form logax=k Exponentiate each side with the base (a) to make the log side equal x. Use the inverse property to find x.

Try these log416 = log216 =

Solving exponential equations In the form ax=k. Take log base a on both sides, to make ax equal x. Use the inverse property to find x.

Try these 4x = 1/16 2x = 16

5.5 Objectives Apply the base properties of logarithms. Use the change of base formula.

Properties of Logarithms loga1=0 and logaa=1 logam+logan=loga(mn) logam-logan=loga(m/n) loga(mr)=rlogam

Change of base formula Let x, a, and b be positive numbers.. ..where a≠ 1 and b≠ 1.

Try these ln 5+ ln 4 log 10-log 5

And more… log 52 log 5 + log 15 – log 10

Assignment Page 430 5, 6, 11, 12 Page 442-443 7-14 31-38