Given that log10 (x-y+1)=0 and 1+log10 (xy)=0, show that x=y=1/sqrt of 10.

Question

ganeshie8 · Answer

change the equations into exponent form and solve

ganeshie8 · Answer

use below :

$\large  \log_a b =  c$  means   $\large   b =  a^c$

anonymous · Answer

do i need put the 1/sqrt of 10 into each of the equation?

ganeshie8 · Answer

nope, start by changing the given two equations to exponent form

ganeshie8 · Answer

$\large  \log_{10} (x-y+1)=0$ means   $\large  x-y + 1 = 10^0$

$\large  x-y + 1 = 1$

$\large  x-y  = 0  ~~~~~~~~~~ \color{Red}{(1)}$

ganeshie8 · Answer

see if you can convert the second log equation ?

ganeshie8 · Answer

Alright, here it is :

$ \large   1+\log_{10} (xy)=0$

$ \large   \log_{10} (xy)=-1$

$ \large   xy =10^{-1}$

$ \large   xy =\dfrac{1}{10}~~~~~~\color{red}{(2)}$

ganeshie8 · Answer

solve both the converted equations   $\color{red}{(1)}$ and  $\color{red}{(2)}$

anonymous · Answer

oh , i got it thanks , but what does you mean solve? just substitute 1/sqrt 10 with x and y?

ganeshie8 · Answer

no, you need to solve both equations and show that x = 1/sqrt(10) and y = 1/sqrt(10)

ganeshie8 · Answer

$\large x - y =  0$

$\large  xy = \frac{1}{10}$

ganeshie8 · Answer

first equation gives you $x = y$
plug that in second equation and solve $y$

anonymous · Answer

ok i know it now , thanks for your guide

ganeshie8 · Answer

good to hear :) can you show me the remaining work ?

anonymous · Answer

from first equation x=y ,
y^2= 1/10
y=1/sqrt (10)
is it correct?

ganeshie8 · Answer

Excellent !!!

anonymous · Answer

really thank you so much

ganeshie8 · Answer

np, you're welcome :)