Question

Which of the following is true regarding the solutions to the logarithmic equation below?

2 log Subscript 6 Baseline (x) = 2. log Subscript 6 Baseline (x squared) = 2. x squared = 6 squared. x squared = 36. x = 6, negative 6.
x = 6 and x = negative 6 are true solutions
x = 6 and x = negative 6 are extraneous solutions
x = 6 is a true solution and x = negative 6 is an extraneous solution
x = 6 is an extraneous solution and x = negative 6 is a true solution

Answer 1

Explanation:

the right answer is

"x = 6 is a true solution, and x = -6 is an extraneous solution."

this is why>

Start with the equation: 2 * log₆(x) = 2.

Divide both sides by 2 to isolate the logarithm: log₆(x) = 1.

Since log₆(6) = 1 (because 6^1 = 6), you can now write x = 6 as a true solution.

Now, for the second equation:

You mentioned: 2 * log₆(x²) = 2.

Divide both sides by 2: log₆(x²) = 1.

Using the properties of logarithms, you can rewrite this as: 2 * log₆(x) = 1.

Divide both sides by 2: log₆(x) = 1/2.

Now, 6^(1/2) = √6, so x = √6 is a valid solution.

However, when you go back to the original equation, x = 6, and plug it into the equation 2 * log₆(x²) = 2:

2 * log₆(6²) = 2 * log₆(36),

2 * log₆(36) ≠ 2.

This means that x = 6 is not a valid solution for the original equation, and it's extraneous.

So, x = 6 is extraneous, and x = -6 is not a valid solution. The only true solution is x = -