Question

solve the equation.
log₈(x² - 7x)=1
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 a comma to s
b. there is no solution.

Explanation:

Step1: Convert to exponential form

By the definition of logarithms, if $\log_{a}b = c$, then $b=a^{c}$. So, $\log_{8}(x^{2}-7x)=1$ becomes $x^{2}-7x = 8^{1}=8$.

Step2: Rearrange to quadratic - form

Rearrange the equation $x^{2}-7x = 8$ to $x^{2}-7x - 8=0$.

Step3: Factor the quadratic equation

Factor $x^{2}-7x - 8$ as $(x - 8)(x+1)=0$.

Step4: Solve for x

Set each factor equal to zero:
If $x - 8=0$, then $x = 8$.
If $x + 1=0$, then $x=-1$.
We need to check for domain. For $\log_{8}(x^{2}-7x)$ to be well - defined, $x^{2}-7x>0$.
When $x = 8$, $x^{2}-7x=8^{2}-7\times8=64 - 56 = 8>0$.
When $x=-1$, $x^{2}-7x=(-1)^{2}-7\times(-1)=1 + 7 = 8>0$.

Answer:

A. $x = 8,-1$