Question

log x + log (x-3) = 1

Answer 1

Given:

The logarithmic equation is:

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

To find:

The value of x.

Solution:

Properties of logarithm used:

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

$\log 10=1$

$\log x$ is defined for $x>0$.

We have,

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

Using properties of logarithm, we get

$\log [x(x-3)]=\log 10$

$x^2-3x=10$

Splitting the middle term, we get

$x^2-5x+2x-10=0$

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

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

$x=5,-2$

In the given equation, we have a term $\log x$. It means the value of x must be greater than 0 or positive. So, the only possible value of x is:

$x=5$

Therefore, the value of x is 5.