Question

If ${\log }_{10}a+{\log }_{10}b={\log }_{10}(a+b)$, then

• A. $a=b=2$
• B. $a=b=1$
• C. $a=\frac{b^2}{1-b}$
• D. $a=\frac{b}{b-1}$

Answer 1

Answer: (A), (D)

The correct options are
A $a=b=2$
D $a=\frac{b}{b-1}$

Given,

${\log }_{10}a+{\log }_{10}b={\log }_{10}(a+b)$

$\therefore {\log }_{10}ab={\log }_{10}(a+b)$ $[\because \log a+\log b=\log ab]$

$\therefore ab=a+b.$

Dividing throughout by $b$ we get

$a=\frac{a}{b}+1\Rightarrow a-\frac{a}{b}=1\Rightarrow a(1-\frac{1}{b})=1\Rightarrow a\left(\frac{b-1}{b}\right)=1\Rightarrow a=\frac{b}{b-1}$

Also, $a=b=2$ satisfies the given equation.

Hence, options A and D are correct.