SOLVING LOGARITHMIC EQUATIONS

Problem 1 :

Solve for x :

½log₉(4) + log₉(2x) = 2log₉(6)

Problem 2 :

Solve for x :

log₂(x) = ½

Solution :

log₂(x) = ½

Convert the above equation to exponential form.

x = 2^½

x = √2

Problem 3 :

Solve for x :

log_1/5(x) = 3

Solution :

Convert the above equation to exponential form. 

x = (1/5)³

x = 1/125

Problem 4 :

Solve for y :

log₃(y) = -2

Solution :

log₃(y) = -2

Convert the above equation to exponential form. 

y = 3^-2

y = 1/3²

y = 1/9

Problem 5 :

Solve for x :

log_x(125√5) = 7

Solution :

log_x(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

√5⁷ = x⁷

Because the exponents are equal, bases can be equated. 

x = √5

Problem 6 :

Solve for x :

log_x(0.001) = -3

Solution :

log_x(0.001) = -3

Convert the above equation to exponential form. 

0.001 = x^-3

1/1000 = 1/x³

Take reciprocal on both sides.

1000 = x³

Take reciprocal on both sides.

1000 = x³

10³ = x³

Because the exponents are equal, bases can be equated. 

10 = x

Problem 7 :

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. 

27^x = 9^-2

(3³)^x = (3²)^-2

3^3x = 3^-4

Because the bases are equal, exponents can be equated. 

3x = -4

Divide both sides by 3.

x = -4/3

Problem 8 :

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 9 :

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)

Since 1/3 ≈ 0.3, we can plug in 1/3 for 0.3 in the above equation.

3x = log₉(1/3)

Solve for x using the properties of logarithm.

3x = log₉(1) - log₉(3)

3x = 0 - log₉(3)

3x = -½log₃3

3x = -½(1)

3x = -½

Multiply both sides by 1/3.

x = -1/6

Problem 10 :

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 11 :

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 12 :

Solve the following equation :

log₂(x) - log₄(x) + log₈(x) = 11/6

Solution :

log₂(x) - log₄(x) + log₈(x) = 11/6

log₂(x) + ½log₂(x) + ⅓log₂(x) = 11/6

Multiply both sides by 6.

6log₂(x) + 3log₂(x) + 2log₂(x) = 11

11log₂(x) = 11

Divide both sides by 11.

log₂(x) = 1

x =  2¹

x = 2

Problem 13 :

Given that

log(x) = m + n

log(y) = m – n

Find the value of log(10x/y²) in terms of m and n.

Solution :

log(10x/y²) :

= log(10x) - log(y²)

= 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 14 :

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).

y = x/(x - 1)

Problem 15 :

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(12/10)

= 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 16 :

Solve for x :

100^√x = log₂(1024)

Solution :

100^√x = log₂(1024)

100^√x = log₂(2¹⁰)

100^√x = 10log₂(2)

(10²)^√x = 10(1)

10^2√x = 10

10^2√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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