Properties of Logarithms

MATH 109 - PrecalculusS. Rook

Overview

• Section 3.3 in the textbook:– Properties of logarithms– Change-of-base formula– Logarithmic scales

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Properties of Logarithms

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Properties of Logarithms• Logarithms can be manipulated using a set of very important

properties:– Product: loga(uv) = logau + logav

• NOTE:– Quotient: loga(u⁄v) = logau – logav

• NOTE:

– Power: loga(un) = n ∙ logau• Applicable to logarithms with ANY valid base including

common and natural logarithms• The bases of the logarithms MUST be the same• Used to write equivalent logarithmic expressions

vuvuv

uaaa

a

a loglogloglog

log

vuvuvu aaaaa logloglogloglog

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Expanding & Compressing Logarithms

• Tips when expanding one logarithm into multiple logarithms with the SAME base as the original:– Work from outer to inner

• Tips when compressing several logarithms of the SAME base into one logarithm of that SAME base:– Apply the power property if necessary

• Removes coefficients from in front of logarithms• Logarithms must NOT have a coefficient in front when combining

– Work from inner to outer– Apply the product and quotient properties of logarithms to

combine

Expanding Logarithms (Example)

Ex 1: Use the properties of logarithms to expand the expression as a sum, difference, and/or constant multiple of logarithms:

a) b)

c)

6

2ln xyz 3

4

5logz

yx

4 23 3log xx

Compressing Logarithms (Example)

Ex 2: Condense the expression to the logarithm of a single quantity:

a) b)

c)

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1ln3ln xx zyx log3log2log

1lnln3ln23

1 2 xxx

Change-of-Base Formula

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Change-of-Base Formula

• Recall last lesson when we discussed that the calculator can only evaluate in base 10 (log) or base e (ln)– Also mentioned that we could “trick” the calculator into

evaluating in other bases

• Change-of-Base Formula:

– Note that the base in the ratios can be any value – just as long as it is the SAME base• e.g.

b

x

b

xxb ln

ln

log

loglog

52ln

32ln

2log

32log32log2

Change-of-Base Formula (Example)

Ex 3: Approximate the logarithm to three decimal places using the change-of-base formula with a) log b) ln:

a)

b)

10

5

2log9

1250log15

Logarithmic Scales

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Logarithmic Scales

• Used to scale very large or very small numbers to a more easily understood interval

• We will see this applied with the Richter Scale

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Richter Scale Magnitude

• The Richter scale is used to convert earthquake intensities to a 0 to 10 scale– A logarithmic scale is required because the intensities can

grow extremely large

• Because intensities are scaled down so compactly, the difference in intensities between any two numbers on the 0 to 10 scale is significant

• Richter Scale Magnitude:

earthquake level zero a ofintensity theis I and intensity,

theis I magnitude, theis M wherelog

0

0

I

IM

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Richter Scale Magnitude (Example)

Ex 4: Compare the intensity of an earthquake that measured 4.5 on the Richter Scale with an earthquake that measured 5.5 on the Richter Scale

Summary

• After studying these slides, you should be able to:– Use the properties of logarithms to condense and expand

logarithmic expressions– Apply the change-of-base formula for bases other than e

or 10– Solve application problems involving logarithmic scales

• Additional Practice– See the list of suggested problems for 3.3

• Next lesson– Exponential & Logarithmic Equations (Section 3.4)

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