Math 1090
Logarithms, Day 2
29 March, 2012
This packet will be integrated with today’s lesson. Throughout class, you will gain
some familiarity with rules for logarithms, the change of base formula, and a deeper
understanding of logarithmic functions in general. Once this packet is complete, you
should have a good set of notes on the second half of the section “Logarithms” in
your textbook.
Name:
Number:
Warm Up: Answer the following questions from yesterday’s lesson.
100, then x .
Graph the function g pxq log4 pxq on the axes below.
1. If 10x
2.
5
3. Convert the equation log4 p2q 1
2
to the equivalent exponential equation.
Part I: Rules for Logarithms
The following rules will help us to solve logarithmic and exponential equations. For the rest
of today’s lesson, a, b, x, y, z P p0, 8q, c, α, γ P R, a 1, and b 1.
Below each rule, write down the corresponding exponential rule.
Rule 1: loga pxy q loga pxq
Exponential: ax
Rule 2: loga
x
y
y
loga py q
ax ay
logapxq logapyq
Exponential:
Rule 3: loga pxc q c loga pxq
Exponential:
Rule 4: loga p1q 0
Exponential:
Now let’s use these rules to solve some problems.
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Use the rules for logarithms to rewrite the following expressions.
4. Write log3 px2 y q as a sum/difference of logarithms.
5. Write log4 ppx
6. Write log5
x
x 1
7. Write log81
1qpxqq as a sum/difference of logarithms.
as a sum/difference of logarithms.
p q as a sum/difference of logarithms.
x5 x 1
x 5
8. Write loge ppabxq2 q as a sum/difference of logarithms.
9. Challenge: Write log4 px2 4q as a sum/difference of logarithms.
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Use the rules for logarithms to rewrite each expression as a single logarithm.
10. log5 pxq
11. log4 px
12. log14 pxq
13. log7 px
log5 py q
1q log4 pxq
log14 py q
1q
2 log7 pxq log7 px 1q
14. Is it possible to write log7 pxq
log6 py q as a single logarithm?
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Part II: Change of Base Formula
The change of base formula allows us to rewrite logarithmic expressions with different bases.
It is useful for calculating approximate values of logarithms.
Change of Base Formula: logx py q loga py q
loga pxq
Rewrite the following expressions as logarithms with base 10. Evaluate with a calculator, if
you have one.
15. log2 p5q
16. loge p15q
17. log100 p1000q
18. log4 p22q
Rewrite the following expressions as logarithms with base e. Evaluate with a calculator, if
you have one.
19. log2 p5q
20. log100 p1000q
21. log4 p22q
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