Problem 1 :
Solve for x :
log2x = 1/2
Solution :
log2x = 1/2
Convert to exponential form.
x = 21/2
x = √2
Problem 2 :
Solve for x :
log1/5x = 3
Solution :
log1/5x = 3
Convert to exponential form.
x = (1/5)3
x = 13/53
x = 1/125
Problem 3 :
Solve for y :
log3y = -2
Solution :
log3y = -2
Convert to exponential form.
y = 3-2
y = 1/32
y = 1/9
Problem 4 :
Solve for x :
logx125√5 = 7
Solution :
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
Problem 5 :
Solve for x :
logx0.001 = -3
Solution :
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
Problem 6 :
Solve for x :
x + 2log279 = 0
Solution :
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
Problem 7 :
If 2logx = 4log3, then find the value of x.
Solution :
2logx = 4log3
Divide each side by 2.
logx = (4log3)/2
logx = 2log3
logx = log32
logx = log9
x = 9
Problem 8 :
If 3x is equal to log(0.3) to the base 9, then find the value of x.
Solution :
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
Problem 9 :
Solve the following equation :
log4(x + 4) + log48 = 2
Solution :
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
Problem 10 :
Solve the following equation :
log6(x + 4) - log6(x - 1) = 1
Solution :
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
Problem 11 :
Solve the following equation :
log2x + log4x + log8x = 11/6
Solution :
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
Problem 12 :
Given that
logx = m + n
logy = m – n
Find the value of log(10x/y2) in terms of m and n.
Solution :
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
Problem 13 :
Given that
logx + logy = log(x + y)
Solve for y in terms of x.
Solution :
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)
Problem 14 :
Given that
log102 = x
log103 = y
Find the value of log101.2 in terms of x and y.
Solution :
= log101.2
= log10(12/10)
= log1012 - log1010
= log10(4 ⋅ 3) - 1
= log104 + log103 - 1
= log1022 + log103 - 1
= 2log102 + log103 - 1
= 2x + y - 1
Problem 15 :
Solve for x :
100√x = log21024
Solution :
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
Kindly mail your feedback to v4formath@gmail.com
We always appreciate your feedback.
©All rights reserved. onlinemath4all.com
Dec 10, 23 10:09 PM
Dec 10, 23 07:42 PM
Dec 10, 23 07:16 PM