Question

prove that 1/log base x xy +1/log base y xy =1​

Answer 1

Let us know logarithmic formulae first:

• $log_{a}(xy)=log_{a}(x)+log_{a}(y)$

• $log_{a}(x/y)=log_{a}(x)-log_{a}(y)$

• $log_{y}(x)=\frac{log_{a}(x)}{log_{a}(y)},\:a>1$

Step-by-step explanation:

Now L.H.S. $=\frac{1}{log_{x}(xy)}+\frac{1}{log_{y}(xy)}$

$=\frac{1}{\frac{log_{a}(xy)}{log_{a}(x)}}+\frac{1}{\frac{log_{a}(xy)}{log_{a}(y)}}$

where $a>1$

• We have used the formula:
• $log_{y}(x)=\frac{log_{a}(x)}{log_{a}(y)},\:a>1$

$=\frac{log_{a}(x)}{log_{a}(xy)}+\frac{log_{a}(y)}{log_{a}(xy)}$

$=\frac{log_{a}(x)+log_{a}(y)}{log_{a}(xy)}$

• Let us use the following formula:
• $log_{a}(xy)=log_{a}(x)+log_{a}(y)$

$=\frac{log_{a}(xy)}{log_{a}(xy)}$

$=\bold{1}$

= R. H. S.

$\Rightarrow \frac{1}{log_{x}(xy)}+\frac{1}{log_{y}(xy)}=1$

Hence proved.