Solving Logarithmic Equations :
In this section, you will learn how to solve logarithmic equations.
To know about logarithm in detail,
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
After having gone through the stuff given above, we hope that the students would have understood how to solve logarithmic equations.
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