Answer

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**Hint:**In this particular question we need to use basic logarithmic properties to simplify the equation. Then we need to further solve the equation and get the desired answer.

**Complete step by step solution:**

In the above question, it is given that,

$\log x + \log (x - 3) = 1$

(Since $\log a + \log b = \log (ab)$ )

Using the above stated property we get,

$ \Rightarrow \log (x(x - 3)) = 1$

Taking antilog on both sides of the equation we get,

$ \Rightarrow anti\log (\log (x(x - 3))) = anti\log 1$

$ \Rightarrow x(x - 3) = {10^1}$

On solving the above equation we get a quadratic equation

$ \Rightarrow {x^2} - 3x = 10$

Subtracting 10 from both sides of the equation

$ \Rightarrow {x^2} - 3x - 10 = 0$

Now solve the above quadratic equation for x

$ \Rightarrow {x^2} - 5x + 2x - 10 = 0$

$ \Rightarrow x(x - 5) + 2(x - 5) = 0$

$ \Rightarrow (x + 2)(x - 5) = 0$

This implies that either $x = - 2$or $x = 5$

**As log can not be a negative value therefore the $x = - 2$ is rejected and hence $x = 5$ is the required solution to the above logarithmic equation.**

**Note:**

Remember to recall the basic logarithmic properties to solve the above question. Note that

$

\log x = 1 \\

\Rightarrow x = {10^1} \\

\Rightarrow x = 10 \\

$

The basic logarithmic algebra includes the following properties:

$

\log a + \log b = \log (ab) \\

\log a - \log b = \log \left( {\dfrac{a}{b}} \right) \\

\log {a^b} = b\log a \\

{\log _a}a = 1 \\

$

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