Problem 1 :
Solve :
log2x = 1/2
Problem 2 :
Solve :
log1/5x = 3
Problem 3 :
Solve :
logx125√5 = 7
Problem 4 :
Solve :
logx0.001 = -3
Problem 5 :
Solve :
log5(5log3x) = 2
Problem 6 :
Solve :
x + 2log279 = 0
Problem 7 :
If 2logx = 4log3, then find the value of x.
Problem 8 :
If 3x is equal to log(0.3) to the base 9, then find the value of x.
Problem 9 :
Solve :
log5 √(7x - 4) - 1/2 = log5 √(x + 2)
Problem 10 :
Solve :
log3x + log9x + log81x = 7/4
Problem 11 :
Solve the following equation :
log4(x + 4) + log48 = 2
Problem 12 :
Solve the following equation :
log6(x + 4) - log6(x - 1) = 1
Problem 13 :
Solve the following equation :
log2x + log4x + log8x = 11/6
Problem 14 :
Given that
logx = m + n
logy = m – n
Find the value of log(10x/y2) in terms of m and n.
Problem 15 :
Given that
logx + logy = log(x + y)
Problem 16 :
Given that
log102 = x
log103 = y
Find the value of log101.2 in terms of x and y.
Problem 17 :
Solve for x :
100√x = log21024
1. Answer :
log2x = 1/2
Convert to exponential form.
x = 21/2
x = √2
2. Answer :
log1/5x = 3
Convert to exponential form.
x = (1/5)3
x = 13/53
x = 1/125
3. Answer :
logx125√5 = 7
Convert to exponential form.
125√5 = x7
5 ⋅ 5 ⋅ 5 ⋅ √5 = x7
Each 5 can be expressed as (√5 ⋅ √5).
Then,
√5 ⋅ √5 ⋅ √5 ⋅ √5 ⋅ √5 ⋅ √5 ⋅ √5 = x7
√57 = x7
Because the exponents are equal, bases can be equated.
x = √5
4. Answer :
logx0.001 = -3
Convert to exponential form.
0.001 = x-3
1/1000 = 1/x3
Take reciprocal on both sides.
1000 = x3
103 = x3
Because the exponents are equal, bases can be equated.
10 = x
5. Answer :
log5(5log3x) = 2
Convert to exponential form.
5log3x = 52
5log3x = 25
Divide each side by 5.
log3x = 5
Convert to exponential form.
x = 35
x = 243
6. Answer :
x + 2log279 = 0
x = -2log279
x = log279-2
Convert to exponential form.
27x = 9-2
(33)x = (32)-2
33x = 3-4
Because the bases are equal, exponents can be equated.
3x = -4
x = -4/3
7. Answer :
2logx = 4log3
Divide each side by 2.
logx = (4log3)/2
logx = 2log3
logx = log32
logx = log9
x = 9
8. Answer :
From the information given, we have
3x = log9(0.3)
Solve for x.
3x = log9(1/3)
3x = log91 - log93
3x = 0 - log93
3x = -log93
3x = -1/log39
3x = -1/log332
3x = -1/2log33
3x = -1/2(1)
3x = -1/2
x = -1/6
9. Answer :
log5√(7x - 4) - 1/2 = log5√(x + 2)
Subtract log5 √(x + 2) from each side.
log5√(7x - 4) - log5√(x + 2) - 1/2 = 0
Add 1/2 to each side.
log5√(7x - 4) - log5√(x + 2) = 1/2
Use quotient rule.
log5[√(7x - 4)/√(x + 2)] = 1/2
Convert to exponential form.
√(7x - 4)/√(x + 2) = 51/2
√(7x - 4)/√(x + 2) = √5
Square each side.
(7x - 4)/(x + 2) = 5
Multiply each side by (x + 2).
7x - 4 = 5(x + 2)
7x - 4 = 5x + 10
Subtract 5x from each side.
2x - 4 = 10
Add 4 to each side.
2x = 14
Divide each side by 2.
x = 7
10. Answer :
log3x + log9x + log81x = 7/4
(1/logx3) + (1/logx9) + (1/logx81) = 7/4
(1/logx3) + (1/logx9) + (1/logx81) = 7/4
(1/logx3) + (1/logx32) + (1/logx34) = 7/4
(1/logx3) + (1/2logx3) + (1/4logx3) = 7/4
(4/4logx3) + (2/4logx3) + (1/4logx3) = 7/4
(4 + 2 + 1)/4logx3 = 7/4
7/4logx3 = 7/4
Multiply each side by 4/7.
1/logx3 = 1
log3x = 1
Convert to exponential form.
x = 31
x = 3
11. Answer :
log4(x + 4) + log48 = 2
Combine the two terms on the left side.
log4[8 ⋅ (x + 4)] = 2
log4(8x + 32) = 2
8x + 32 = 42
8x + 32 = 16
Subtract by 32 from both sides
8x = -16
Divide both sides by 8.
x = -2
12. Answer :
log6(x + 4) - log6(x - 1) = 1
Combine the two terms on the left side
log6[(x + 4)/(x - 1)] = 1
(x + 4)/(x - 1) = 61
(x + 4)/(x - 1) = 6
x + 4 = 6(x - 1)
x + 4 = 6x - 6
Subtract 6x from both sides.
x - 6x + 4 = -6
-5x + 4 = -6
Subtract 4 from both sides.
-5x = -6 - 4
-5x = -10
Divide both sides by -5.
x = 2
13. Answer :
log2x + log4x + log8x = 11/6
(1/logx2) + (1/logx4) + (1/logx8) = 11/6
(1/logx2) + (1/logx22) + (1/logx23) = 11/6
(1/logx2) + (1/2logx2) + (1/3logx2) = 11/6
(1/logx2) (1 + 1/2 + 1/3) = 11/6
(1/logx2)(11/6) = 11/6
1/logx2 = 1
1 = logx2
x = 2
14. Answer :
log(10x/y2) = log10x - 1ogy2
= log10 + logx - 2logy
= 1 + logx - 2logy
Substitute.
= 1 + (m + n) - 2(m - n)
= 1 + m + n - 2m + 2n
= 1 - m + 3n
15. Answer :
logx + logy = log(x + y)
Use the Product Rule of Logarithm on the left side.
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)
16. Answer :
= log101.2
= log10(12/10)
= log1012 - log1010
= log10(4 ⋅ 3) - 1
= log104 + log103 - 1
= log1022 + log103 - 1
= 2log102 + log103 - 1
= 2x + y - 1
17. Answer :
100√x = log21024
100√x = log2210
100√x = 10log22
100√x = 10(1)
100√x = 10
(102)√x = 10
102√x = 10
102√x = 101
2√x = 1
Divide both sides by 2.
√x = 1/2
Square both sides.
(√x)2 = (1/2)2
x = 1/4
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