Question

The logarithm of a product is the sum of the logarithms of the factors. An exponent, pp, signifies that a number is being multiplied by itself pp number of times. Because the logarithm of a product is the sum of the logarithms of the factors, the logarithm of a number, $x$, to an exponent, $p$, is the same as the logarithm of $x$ added together $p$ times, so it is equal to $p{\log }_b(x)$. A useful property of logarithms states that the logarithm of a product of two quantities is the sum of the logarithms of the two factors. In symbols, ${\log }_b(xy)={\log }_b(x)+{\log }_b(y)$.
If $\log \frac{a}{b}+\log \frac{b}{a}=\log (a+b)$ then, which of the following is true?

• A. $a+b=0$
• B. $b-a=1$
• C. $a^2=1-b$
• D. $b^2=1-a$

Answer 1

Answer: (A)

Given: $\log \frac{a}{b}+\log \frac{b}{a}=\log (a+b)$
$\Rightarrow \log \left(\frac{a}{b}\times \frac{b}{a}\right)=\log (a+b)$
$\left\{\because {\log }_m\left((ab)={\log }_{m}a+{\log }_{m}b\right)\right\}$
$\Rightarrow \log (1)=\log (a+b)\{\because \log 1=0\}$
$\Rightarrow \log (a+b)=0$