solve for $x$.\n$log_{4}(x^{2}-76) + log_{4}(5) = log_{4}(25)$\nif no solution exists, enter none. if…

solve for $x$.\n$log_{4}(x^{2}-76) + log_{4}(5) = log_{4}(25)$\nif no solution exists, enter none. if multiple solutions exist, separate each with a comma.

solve for $x$.\n$log_{4}(x^{2}-76) + log_{4}(5) = log_{4}(25)$\nif no solution exists, enter none. if multiple solutions exist, separate each with a comma.

Answer

Explanation:

Step1: Combine left-hand side logs

Use log product rule: $\log_b(a) + \log_b(c) = \log_b(ac)$ $\log_4\left(5(x^2 - 76)\right) = \log_4(25)$

Step2: Eliminate equal base logs

If $\log_b(A)=\log_b(B)$, then $A=B$ $5(x^2 - 76) = 25$

Step3: Isolate the quadratic term

Divide both sides by 5, then add 76 $x^2 - 76 = 5$ $x^2 = 5 + 76 = 81$

Step4: Solve for x, check domain

Take square root; ensure $x^2-76>0$ $x = \pm\sqrt{81} = \pm9$ For $x=9$: $9^2-76=81-76=5>0$; for $x=-9$: $(-9)^2-76=5>0$, both valid.

Answer:

9, -9