1. Find the logarithm of 32 to the base 2.
2. Find the value of log2√31728.
3. Find the value of log(0.0001) to the base 0.1.
4. Find the value of log (1/81) to the base 9.
5. Find the value of log(0.0625) to the base 2.
6. Find the value of log(0.3) to the base 9.
7. If loga(√2) = 1/6, find the value of a.
8. Simplify : (1/2)log1025 - 2log103 + log1018.
9. Given log2 = 0.3010 and log3 = 0.4771, find the value of log6.
10. If 2logx = 4log3, then find the value of x.
11. If logabc = x, logbca = y and logcab = z, then find the value of
1/(x + 1) + 1/(y + 1) + 1/(z + 1)
12. If a = log2412, b = log3624 and c = log4836, then find the value of (1 + abc) in terms of b and c.
13. If logx + logy = log(x + y), solve for y in terms of x.
1. Answer :
log232 = log2(2)5
= 5log2(2)
= 5(1)
= 5
2. Answer :
Write 1728 as a power of 2√3.
1728 = 2 x 2 x 2 x 2 x 2 x 2 x 3 x 3 x 3
1728 = 26 x 33
1728 = 26 x [(√3)2]3
1728 = 26 x (√3)6
1728 = (2√3)6
log2√3(1728) = log2√3(2√3)6
Using the power rule of logarithms,
= 6log2√3(2√3)
= 6(1)
= 6
3. Answer :
log0.10.0001 = log0.1(0.1)4
= 4log0.10.1
= 4(1)
= 4
4. Answer :
log9(1/81) = log91 - log981
= 0 - log9(9)2
= -2log99
= -2(1)
= -2
5. Answer :
log2(0.0625) = log2(0.5)4
= 4log2(0.5)
= 4log2(1/2)
= 4(log21 - log22)
= 4(0 - 1)
= 4(-1)
= -4
6. Answer :
log9(0.3) = log9(1/3)
= log91 - log93
= 0 - log93
= -log93
= -1 / log39
= -1 / log332
= -1 / 2log33
= -1 / 2(1)
= -1/2
7. Answer :
loga(√2) = 1/6
Write the equation in exponential form.
√2 = a1/6
Raise to the power 6 on both sides.
(√2)6 = (a1/6)6
(21/2)6 = a
23 = a
8 = a
8. Answer :
= (1/2)log1025 - 2log103 + log1018
Using power rule of logarithms,
= log10251/2 - log1032 + log1018
= log10(52)1/2 - log1032 + log1018
= log105 - log109 + log1018
= log105 + log1018 - log109
Using the product rule of logarithms,
= log10(5 x 18) - log109
= log1090 - log109
Using the quotient rule of logarithms,
= log10(90/9)
= log1010
= 1
9. Answer :
log6 = log(2 ⋅ 3)
= log2 + log3
Substitute the values of log2 and log3.
= 0.3010 + 0.4771
= 0.7781
10. Answer :
2logx = 4log3
Divide each side by 2.
logx = 2log3
logx = log32
logx = log9
x = 9
11. Answer :
x + 1 = logabc + logaa = logaabc
y + 1 = logbca + logbc = logbabc
z + 1 = logcab + logcc = logcabc
1/(x + 1) = 1/logaabc = logabca
1/(y + 1) = 1/logbabc = logabcb
1/(z + 1) = 1/logcabc = logabcc
1/(x + 1) + 1/(y + 1) + 1/(z + 1) = logabca + logabcb + logabcc
= logabcabc
= 1
12. Answer :
1 + abc = 1 + log2412 ⋅ log3624 ⋅ log4836
= 1 + log3612 ⋅ log4836
= 1 + log4812
= log4848 + log4812
= log48(48 ⋅ 12)
= log48(2 ⋅ 12)2
= 2log4824
= 2log3624 ⋅ log4836
= 2bc
13. Answer :
logx + logy = log(x + y)
Using the product rule of logarithms on the left side of the equation,
log(xy) = log(x + y)
xy = x + y
Subtract y from both sides.
xy - y = x
Factor.
y(x - 1) = x
Divide both sides by (x - 1).
y = x/(x - 1)
Fundamental Laws of Logarithms
Difference between Bar Value and Negative Value
Multiplication of Two Logarithms
Relationship between Exponents and Logarithms
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