Solving Logarithmic Equations

Which of the following is the solution to the equation log₃(x) = 4?

• 9

• 27

• 81

• 243

Answer explanation

Using the definition of logarithms, we can rewrite the equation as 3⁴ = x, which simplifies to 81 = x. Therefore, the correct answer is c) x = 81.

What is the solution to the equation log₅(2x + 1) = 2?

• 3

• 4

• 5

• 6

Answer explanation

Rewriting the equation in exponential form, we get 5² = 2x + 1. Solving for x, we subtract 1 from both sides and then divide by 2 to obtain x = 12/2 = 6. Therefore, the correct answer is d) x = 6.

Which value of x satisfies the equation log₄(x + 3) = log₄(8)?

• 2

• 3

• 5

• 8

Answer explanation

Since both sides of the equation have the same base (4), we can equate the arguments: x + 3 = 8. Solving for x, we subtract 3 from both sides to get x = 5. Therefore, the correct answer is c) x = 5.

What is the solution to the equation ln(2x) = 3?

• 2.718

• 4.481

• 6.049

• 10.852

Answer explanation

To solve this equation, we need to remember that ln represents the natural logarithm with base e. Rewriting the equation in exponential form, we have e³ = 2x. Dividing both sides by 2, we find x = e³/2. Therefore, the correct answer is d) x ≈ 10.852.

Which of the following is a solution to the equation log₂(x - 1) + log₂(x + 2) = 3?

• 5

• 3

• 1

• -4

Answer explanation

Using the logarithmic property logₐ(m) + logₐ(n) = logₐ(m * n), we can combine the logarithms: log₂((x - 1)(x + 2)) = 3. This simplifies to log₂(x² + x - 2) = 3. Writing the equation in exponential form, we have 2³ = x² + x - 2. Simplifying further, we get x² + x - 2 = 8. Rearranging and solving the quadratic equation, we find x = -4 or x = 3. Therefore, the correct answer is c) x = 3.

What is the solution to the equation logₓ(8x) - logₓ(2x) = 1?

• 4

• 2

• 3

• 5

Answer explanation

Applying the logarithmic property logₐ(m) - logₐ(n) = logₐ(m/n), we can simplify the equation to logₓ(8x/2x) = 1. This further simplifies to logₓ(4) = 1. Rewriting in exponential form, we have x¹ = 4. Therefore, the correct answer is c) x = 4.

What value of x satisfies the equation log(5x) + log(2x) = log(40)?

• -2

• 1

• 2

• 3

Answer explanation

Using the logarithmic property log(m) + log(n) = log(m n), we can combine the logarithms: log(5x 2x) = log(40). Simplifying further, we have log(10x²) = log(40). Since the logarithm function is one-to-one, we equate the arguments: 10x² = 40. Dividing both sides by 10, we find x² = 4. Taking the square root of both sides, we get x = ±2. Therefore, the correct answer is c) x = 2.

Which value of x satisfies the equation log₄(x - 1) - log₄(2) = 2?

• 10

• 17

• 32

• 33

Answer explanation

Using the logarithmic property logₐ(m) - logₐ(n) = logₐ(m/n), we can simplify the equation to log₄((x - 1)/2) = 2. Writing in exponential form, we have 4² = (x - 1)/2. Simplifying further, we get 16 = (x - 1)/2. Multiplying both sides by 2, we find 32 = x - 1. Adding 1 to both sides, the solution is x = 33. Therefore, the correct answer is d) x = 33.

What is the solution to the equation log₇(x - 4) = log₇(x - 5) + log₇(2)?

• 13

• 10

• 12

• 15

Answer explanation

Applying the logarithmic property logₐ(m) + logₐ(n) = logₐ(m n), we can simplify the equation to log₇((x - 5) 2) = log₇(x - 4). Since the logarithm function is one-to-one, we equate the arguments: (x - 5) * 2 = x - 4. Solving for x, we find x = 13. Therefore, the correct answer is c) x = 13.

Which of the following is a solution to the equation log(3x) - log(2x) = 1?

• 2

• 3

• 4

• No solution

Answer explanation

Using the logarithmic property log(m) - log(n) = log(m/n), we can simplify the equation to log((3x)/(2x)) = 1. Simplifying further, we get log(3/2) = 1. Since log(3/2) is not equal to 1, there is no solution to this equation. Therefore, the correct answer is d) There is no solution.