Question

what is the solution of \\(\log(t - 3) = \log(17 - 4t)\\)?
\\(\circ\\) 4
\\(\circ\\) 5
\\(\circ\\) 15
\\(\circ\\) 20

Explanation:

Step1: Use the property of logarithms

If $\log(a)=\log(b)$, then $a = b$ (for the same base and valid arguments). So we set $t - 3=17 - 4t$.

Step2: Solve the linear equation

Add $4t$ to both sides: $t+4t - 3=17 - 4t+4t$, which simplifies to $5t - 3 = 17$.
Add 3 to both sides: $5t-3 + 3=17 + 3$, so $5t=20$.
Divide both sides by 5: $t=\frac{20}{5}=4$.
We should also check the domain: For $\log(t - 3)$, $t-3>0\Rightarrow t > 3$. For $\log(17 - 4t)$, $17-4t>0\Rightarrow 4t<17\Rightarrow t<\frac{17}{4} = 4.25$. When $t = 4$, $4>3$ and $4<4.25$, so it is valid.

Answer:

4 (the option corresponding to 4)