3.4 Properties of Logarithmic Functions

Properties of Logarithms

Product Rule: logb(RS) = logbR + logbS

Ex) ln 8x =

Quotient Rule: logb(R/S) = logbR – logbS

Ex) log (3/x) =

Power Rule: logbRc = c logbR

Ex) log2x-2 =

You Try! Write the expression as a sum or difference of logarithms

or multiples of logarithms

1. ln 9y

2. log2y5

3. ln (x2/y3)

You Try! Write the expression as a single logarithm

log5x + log5y

2 ln x + 3 ln y

4 log (xy) – 3log (yz)

Change-of-Base Formula

For positive real numbers a, b, and x with

a ≠1 and b ≠1,

or

logb x =logxlogb

logb x =lnxlnb

Use the change-of-base formula and your calculator toevaluate the logarithm:

Log27

You Try!Log8175

Log0.512

Graphing Logarithmic functions

• If b > 1, the graph of g(x) = logbx is a vertical stretch or shrink of the graph of ln (x) by the factor of 1/ ln b.

•If 0 < b < 1, a reflection across the x-axis is required as well.

g(x)=logb x=lnxlnb

=1

lnblnx

Describe how to transform the graph of f(x) = ln x into the graph of the given function:

g(x) = log5x

Describe how to transform the graph of f(x) = ln x into the graph of the given function:

h(x) = log1/4x

f(x) = ln (x3)

DomainRangeContinuity IncreasingDecreasingAsymptotesEnd Behavior

Homework

Pg. 317 (4, 6, 12, 20, 22, 28, 40, 42, 49)