Question

solve the equation. log _4(x^2 - 3x)=1 select the correct choice below and fill in any ar a. x = (simplify your answer, including any radi 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, $\log_{4}(x^{2}-3x)=1$ becomes $x^{2}-3x = 4^{1}=4$.

Step2: Rearrange to quadratic - form

Rearrange the equation $x^{2}-3x = 4$ to $x^{2}-3x - 4=0$.

Step3: Factor the quadratic equation

Factor $x^{2}-3x - 4$ as $(x - 4)(x+1)=0$.

Step4: Solve for x

Set each factor equal to zero: $x - 4=0$ gives $x = 4$; $x + 1=0$ gives $x=-1$.

Step5: Check for domain

For $\log_{4}(x^{2}-3x)$ to be well - defined, $x^{2}-3x>0$.
When $x = 4$, $x^{2}-3x=4^{2}-3\times4=16 - 12 = 4>0$.
When $x=-1$, $x^{2}-3x=(-1)^{2}-3\times(-1)=1 + 3=4>0$.

Answer:

A. $x = 4,-1$