Question

what is the solution to \\(\log_{7}(x - 4)=\log_{7}(4x + 5)\\)?\\(x = -3\\)\\(x = -2\\)\\(x = -1\\)\\(x = 0\\)there is no solution.

Explanation:

Step1: Use the property of logarithms

If $\log_b a=\log_b c$, then $a = c$ (for $b>0,b
eq1,a>0,c>0$). So we set $x - 4=4x + 5$.

Step2: Solve the equation

Subtract $x$ from both sides: $- 4 = 3x+5$.
Subtract 5 from both sides: $-9 = 3x$.
Divide by 3: $x=-3$.

Step3: Check the domain

For $\log_7(x - 4)$ to be defined, $x-4>0\Rightarrow x > 4$. For $\log_7(4x + 5)$ to be defined, $4x + 5>0\Rightarrow x>-\frac{5}{4}$. But our solution $x = - 3$ does not satisfy $x>4$, so we discard it.
We check other options too. For $x=-2$: $x - 4=-6<0$ (invalid). For $x=-1$: $x - 4=-5<0$ (invalid). For $x = 0$: $x - 4=-4<0$ (invalid). So no solution.

Answer:

There is no solution.