Question

solve the equation.
log₈x + log₈(x - 12)=2

select the correct choice below and fill in any answer boxes present in your choice.

a. x=
(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression. use

b. there is no solution.

Explanation:

Step1: Apply log - product rule

By the rule $\log_aM+\log_aN = \log_a(MN)$, we have $\log_8[x(x - 12)]=2$.

Step2: Convert to exponential form

The logarithmic equation $\log_8[x(x - 12)]=2$ can be rewritten in exponential form as $8^2=x(x - 12)$ since $\log_a b=c$ is equivalent to $a^c = b$. So, $64=x^2-12x$.

Step3: Rearrange to quadratic form

Rearrange the equation to get $x^2-12x - 64=0$.

Step4: Factor the quadratic equation

Factor $x^2-12x - 64$ as $(x - 16)(x + 4)=0$.

Step5: Solve for x

Set each factor equal to zero: $x-16 = 0$ gives $x = 16$; $x + 4=0$ gives $x=-4$.

Step6: Check for domain

For $\log_8x$ and $\log_8(x - 12)$ to be well - defined, $x>0$ and $x-12>0$ (i.e., $x>12$). So, $x=-4$ is an extraneous solution.

Answer:

A. $x = 16$