Question

solve log 3x = log 5 + log (x - 2)
give exact answer
give solution set

Explanation:

Step1: Apply log product rule

Using the property $\log a + \log b = \log(ab)$, we rewrite the right - hand side of the equation $\log 3x=\log 5+\log(x - 2)$ as $\log 3x=\log[5(x - 2)]$.

Step2: Use one - to - one property of logs

If $\log a=\log b$ (assuming the same base for the logarithms), then $a = b$. So we have the equation $3x=5(x - 2)$.

Step3: Solve the linear equation

Expand the right - hand side: $3x = 5x-10$.
Subtract $3x$ from both sides: $0 = 5x-3x - 10$, which simplifies to $0 = 2x-10$.
Add 10 to both sides: $2x=10$.
Divide both sides by 2: $x = 5$.

Step4: Check the domain

For the logarithm $\log 3x$ to be defined, $3x>0\Rightarrow x > 0$.
For the logarithm $\log(x - 2)$ to be defined, $x-2>0\Rightarrow x>2$.
When $x = 5$, $3x=15>0$ and $x - 2=3>0$, so $x = 5$ is in the domain.

Answer:

The exact answer is $x = 5$ and the solution set is $\{5\}$.