4'4 RULES

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4.4 RULES OF LOGARITHMS
Mrs. J. Kessler
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RULES OF LOGARITHMS
Let M, N, and a be positive real numbers with a ?
1, and let r be any real number.
The logarithm of the product of two (or more)
numbers is the sum of the logarithms of the
numbers.
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RULES OF LOGARITHMS
Let M, N, and a be positive real numbers with a ?
1, and let r be any real number.
The logarithm of the quotient of two (or more)
numbers is the difference of the logarithms of
the numbers.
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RULES OF LOGARITHMS
Let M, N, and a be positive real numbers with a ?
1, and let r be any real number.
The logarithm of a number to the power r is r
times the logarithm of the number.
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Rules of Logarithms Summary
For x gt 0, y gt 0, a gt 0, a ? 1, and any real
number r
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Rules Of Logarithms Examples
log35(x-3)
log35 log3(x-3)
log x2y2
log x2 log y2
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Rules Of Logarithms Examples

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Rules Of Logarithms Examples

• log3x5

5log3x
log (x3y2z5)
3 log x 2 log y 5 logz
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Writing Expressions In Expanded Form
Write the expression in expanded form.
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Writing Expressions In Expanded Form
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Writing Expressions In Expanded Form
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Evaluating Using the Rules Of Logarithms
Given that log 5 z 3 and log 5 y 2, evaluate
each expression.
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Day 2

• Evaluating, Condensing,
• Change of base rule

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Evaluating Using the Rules Of Logarithms
Given that log 5 z 3 and log 5 y 2, evaluate
each expression.
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Evaluating Using the Rules Of Logarithms
Given that log 5 z 3 and log 5 y 2, evaluate
each expression.
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Evaluating Using the Rules Of Logarithms
Given that log 5 z 3 and log 5 y 2, evaluate
each expression.
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Writing Log Expressions in Condensed Form
Write the expression in condensed form.
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Writing Log Expressions in Condensed Form
Write the expression in condensed form.
ln
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Writing Log Expressions in Condensed Form
Write the expression in condensed form.
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Writing Log Expressions in Condensed Form
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Change-of-base Formula
Let a, b, and c be positive real numbers with a ?
1 and b ? 1. Then logb x can be converted to a
different base as follows
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Using a Change of Base to Compute Logarithms
Compute log513 by changing to (a) common
logarithms and (b) natural logarithms.
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Using Change of Base to graph
Graph y log3x
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Logarithms With Exponential Base
Heres an example where k 1.
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Examples
Evaluating an Expression
Find the value of each expression without using a
calculator.
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Evaluating an Expression
New Rule
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Evaluating an Expression
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4.4 Day 3Exponential and logarithmic modeling
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Matching Data to an Exponential Curve
Find the exponential function of the form f (x)
aekx that passes through the points (0, 2) and
(3, 8).
Substitute (0, 2) into f (x) aekx.
k(0)
So a 2 and f (x) 2ekx . Now substitute (3,
8) in to the equation.
(3k)
k(3)
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Matching Data to an Exponential Curve
Now solve for k.
3k
(3k)
k
k
f (x) aekx
Thus,
is the desired function.
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Matching Data to an Exponential Curve

• Could we do this problem with regression?

We need y aekx
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Expressing an Exponential Model in Base e.

• y abx
• is equivalent to
• y ae(lnb)x

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Expressing an Exponential Model in Base e.

• y abx y ae(lnb)x

y 2e((ln1.587401052)x)
y 2e.4620981204x
Graph to check.
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Find an exponential model

• Find the exponential model for the following data

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Exponential modeling
y 2.64e((ln1.482728676)x)
y 2.64e .39388409x
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More modeling
Plot the following
Does this look exponential?
Change window so that y min 0