quiz 3.2. 3.3 properties of logarithms date: ____________
TRANSCRIPT
Quiz 3.2
• Write the log as an exponent. • Evaluate the log without using your calculator. 4
3.3Properties of Logarithms
Date: ____________
Change of Base Theorem
logax = log x log a
logax = ln x ln a
OR
1) log4 9
2) log7 3
Write the logarithm in terms of
common logarithms.
log 9
log 4
log 3
log 7
1) log5 8
2) log2 7
Write the logarithm in terms of
natural logarithms.
ln 8
ln 5
ln 7
ln 2
Evaluating Logarithms
log523 = log23 log5 ≈ 1.948
OR
log523 = ln23 ln5 ≈ 1.948
Other Examples
log3149 = log149 log3 ≈ 4.555
log7300 = log300 log7 ≈ 2.9312
Properties of Logarithms
Product Property
Quotient Property
Power Property
Let b, u, and v be positive numbers such that b ≠ 1.
logbuv = logbu + logbv
logbun = nlogbu
logb = logbu − logbvuv
1) log9
4
7
2) log9 28
Use log9 7 0.8856 and log9 4 0.6309
to approximate the following.
0.2547
0.6309 0.8856 1.5165
3)log9 256 4 0.6309 2.5236
log9 4 log9 7 0.6309 0.8856
9log (4 7) log9 4 log9 7
49log 4 4log9 4
1) log3
2
7
2) log314
Use log32 0.6310 and log3 7 1.7712
to approximate the following.
1.1402
0.6310 1.7712 2.4022
3) log3128 7 0.6310 4.4170
log32 log3 7 0.6310 1.7712
3log (2 7) log32 log3 7
73log 2 7log32
Find the exact value of the logarithm.
48log 8
14
8log 81
4
4 3ln e34ln e
3
4
31
log81
13log 81 31 log 81
43log 3 4
Use the properties of logarithms to simplify the given expression.
51
log15
5 5log 1 log 15
50 log 15
5 5log 5 log 3
51 log 3
51 log 3
5log (5 3)
Use the properties of logarithms to simplify the given expression.
31
log18
3 3log 1 log 18
30 log 18
3 3log 9 log 2 2
3 3log 3 log 2 32 log 2
3log 9 2
Use the properties of logarithms to simplify the given expression.
3log 3012
3log (30)1
32log 30
13 32
log 3 log 10
132
1 log 10 1
32log 10
132
log 3 10
Write as the sum, difference, or product of logarithms. Simplify, if possible.
log4
4x6
y = log44x6 – log4 y
= log44 + log4x6 – log4 y
= log44 + 6log4x – log4 y
= 1 + 6log4x – log4 y
Write as the sum, difference, or product of logarithms. Simplify, if possible.
log7
6x3y = log76
– log7 x3y
= log76 −(log7x3 + log7 y)
= log76 − (3log7x + log7 y)
= log76 − 3log7x − log7 y
Write as the sum, difference, or product of logarithms. Simplify, if possible.
log57 x
= log57 + log5 x
= log57 + log5x½
= log57 + ½log5x
Write as the sum, difference, or product of logarithms. Simplify, if possible.
loga
x3y5
z = loga
x3y5
z
½
= ½loga
x3y5
z
= ½(loga x3+ loga y
5 – logaz)
= ½(3loga x+ 5loga y – logaz)
Condense the expression to the logarithm of a single quantity.
log35 + 6log3x − log3 7
= log35 + log3x6 − log3 7
= log3(5x6) − log3 7
= log3 7 5x6
Condense the expression to the logarithm of a single quantity.
3log8 x − 5log8 y + log8 15
= log8x3 − log8 y
5 + log8 15
•15 log8y5
= x3
= log8y5
15x3
x3
Condense the expression to the logarithm of a single quantity.
3log8 x − 5log8 y − log8 15
= log8x3 − log8 y
5 − log8 15
= log815y5