# SOLVING LOGARITHMIC EQUATIONS

Problem 1 :

Solve for x :

Solution :

Convert the above equation to exponential form.

Problem 2 :

Solve for x :

Solution :

Convert the above equation to exponential form.

Problem 3 :

Solve for y :

log3(y) = -2

Solution :

log3(y) = -2

Convert the above equation to exponential form.

y = 3-2

Problem 4 :

Solve for x :

logx(125√5) = 7

Solution :

logx(125√5) = 7

Convert the above equation 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

√5= x7

Because the exponents are equal, bases can be equated.

x = √5

Problem 5 :

Solve for x :

logx(0.001) = -3

Solution :

logx(0.001) = -3

Convert the above equation to exponential form.

0.001 = x-3

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 + 2log27(9) = 0

Solution :

x + 2log27(9) = 0

x = -2log27(9)

x = log27(9)-2

Convert the above equation to exponential form.

27x = 9-2

(33)= (32)-2

33x = 3-4

Because the bases are equal, exponents can be equated.

3x = -4

Problem 7 :

If 2logx = 4log3,  then find the value of x

Solution :

2log(x) = 4log(3)

Divide both sides by 2.

log(x) = 2log(3)

log(x) = log(32)

log(x) = log(9)

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) - log9(3)

3x = 0 - log9(3)

3x = -log93

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.

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 :

Solution :

log2(x) = 1

x = 21

x = 2

Problem 12 :

Given that

log(x) = m + n

log(y) = m – n

Solution :

= log(10) + log(x) - 2log(y)

= 1 + logx - 2logy

Substitute.

= 1 + (m + n) - 2(m - n)

= 1 + m + n - 2m + 2n

= 1 - m + 3n

Problem 13 :

Given that

log(x) + log(y) = 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).

Problem 14 :

Given that

log10(2) = x

log10(3) = y

Find the value of log10(1.2) in terms of x and y.

Solution :

= log10(1.2)

= log10(12) - log10(10)

= log10(4 ⋅ 3) - 1

= log10(4) + log10(3) +  - 1

= log10(22) + log10(3) +  - 1

= 2log10(2) + log10(3) +  - 1

Substitute.

= 2x + y - 1

Problem 15 :

Solve for x :

100x = log2(1024)

Solution :

100x = log2(1024)

100x = log2(210)

100= 10log2(2)

(102)= 10(1)

102= 10

102= 101

2√x = 1

Divide both sides by 2.

Square both sides.

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