Question

d. \\(\log (5m + 1) = \log (m^2 - 49)\\)

Explanation:

Step1: Set arguments equal (log property)

If $\log(a) = \log(b)$, then $a = b$ (for valid domains).
$5m + 1 = m^2 - 49$

Step2: Rearrange to quadratic equation

Bring all terms to one side.
$m^2 - 5m - 50 = 0$

Step3: Factor the quadratic

Find two terms that multiply to $-50$ and add to $-5$.
$(m - 10)(m + 5) = 0$

Step4: Solve for m

Set each factor equal to 0.
$m - 10 = 0 \implies m = 10$; $m + 5 = 0 \implies m = -5$

Step5: Check domain validity

Log arguments must be positive:
For $m=10$: $5(10)+1=51>0$, $10^2-49=51>0$ (valid)
For $m=-5$: $5(-5)+1=-24<0$ (invalid, discard)

Answer:

$m = 10$