1. Find the logarithm of 32 to the base 2.
2. Find the value of log_{2}_{√3}1728.
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 log_{a}(√2) = 1/6, find the value of a.
8. Simplify : (1/2)log_{10}25 - 2log_{10}3 + log_{10}18.
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 log_{a}bc = x, log_{b}ca = y and log_{c}ab = z, then find the value of
1/(x + 1) + 1/(y + 1) + 1/(z + 1)
12. If a = log_{24}12, b = log_{36}24 and c = log_{48}36, 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 :
log_{2}32 = log_{2}(2)^{5}
= 5log_{2}(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 = 2^{6} x 3^{3}
1728 = 2^{6} x [(√3)^{2}]^{3}
1728 = 2^{6} x (√3)^{6}
1728 = (2√3)^{6}
log_{2}_{√3}(1728) = log_{2}_{√3}(2√3)^{6}
Using the power rule of logarithms,
= 6log_{2}_{√3}(2√3)
= 6(1)
= 6
3. Answer :
log_{0.1}0.0001 = log_{0.1}(0.1)^{4}
= 4log_{0.1}0.1
= 4(1)
= 4
4. Answer :
log_{9}(1/81) = log_{9}1 - log_{9}81
= 0 - log_{9}(9)^{2}
= -2log_{9}9
= -2(1)
= -2
5. Answer :
log_{2}(0.0625) = log_{2}(0.5)^{4}
= 4log_{2}(0.5)
= 4log_{2}(1/2)
= 4(log_{2}1 - log_{2}2)
= 4(0 - 1)
= 4(-1)
= -4
6. Answer :
log_{9}(0.3) = log_{9}(1/3)
= log_{9}1 - log_{9}3
= 0 - log_{9}3
= -log_{9}3
= -1 / log_{3}9
= -1 / log_{3}3^{2}
= -1 / 2log_{3}3
= -1 / 2(1)
= -1/2
7. Answer :
log_{a}(√2) = 1/6
Write the equation in exponential form.
√2 = a^{1/6}
Raise to the power 6 on both sides.
(√2)^{6} = (a^{1/6})^{6}
(2^{1/2})^{6} = a
2^{3} = a
8 = a
8. Answer :
= (1/2)log_{10}25 - 2log_{10}3 + log_{10}18
Using power rule of logarithms,
= log_{10}25^{1/2} - log_{10}3^{2} + log_{10}18
= log_{10}(5^{2})^{1/2} - log_{10}3^{2} + log_{10}18
= log_{10}5 - log_{10}9 + log_{10}18
= log_{10}5 + log_{10}18 - log_{10}9
Using the product rule of logarithms,
= log_{10}(5 x 18) - log_{10}9
= log_{10}90 - log_{10}9
Using the quotient rule of logarithms,
= log_{10}(90/9)
= log_{10}10
= 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 = log3^{2}
logx = log9
x = 9
11. Answer :
x + 1 = log_{a}bc + log_{a}a = log_{a}abc
y + 1 = log_{b}ca + log_{b}c = log_{b}abc
z + 1 = log_{c}ab + log_{c}c = log_{c}abc
1/(x + 1) = 1/log_{a}abc = log_{abc}a
1/(y + 1) = 1/log_{b}abc = log_{abc}b
1/(z + 1) = 1/log_{c}abc = log_{abc}c
1/(x + 1) + 1/(y + 1) + 1/(z + 1) = log_{abc}a + log_{abc}b + log_{abc}c
= log_{abc}abc
= 1
12. Answer :
1 + abc = 1 + log_{24}12 ⋅ log_{36}24 ⋅ log_{48}36
= 1 + log_{36}12 ⋅ log_{48}36
= 1 + log_{48}12
= log_{48}48 + log_{48}12
= log_{48}(48 ⋅ 12)
= log_{48}(2 ⋅ 12)^{2}
= 2log_{48}24
= 2log_{36}24 ⋅ log_{48}36
= 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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