Solve for x :
1) log2x = 1/2
2) log1/5x = 3
3) logx125√5 = 7
4) logx0.001 = -3
5) log5(5log3x) = 2
6) x + 2log279 = 0
7) If 2logx = 4log3, then find the value of x.
8) If 3x is equal to log(0.3) to the base 9, then find the value of x.
Solve for x :
9) log5 √(7x - 4) - 1/2 = log5 √(x + 2)
Solve for x :
10) log3x + log9x + log81x = 7/4
Answer (1) :
log2x = 1/2
Convert to exponential form.
x = 21/2
x = √2
Answer (2) :
log1/5x = 3
Convert to exponential form.
x = (1/5)3
x = 13/53
x = 1/125
Answer (3) :
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
Answer (4) :
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
Answer (5) :
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
Answer (6) :
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
Answer (7) :
2logx = 4log3
Divide each side by 2.
logx = (4log3) / 2
logx = 2log3
logx = log32
logx = log9
x = 9
Answer (8) :
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
Answer (9) :
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
Answer (10) :
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
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