# What is the answer for x and y lg(xy)=9 lgx=3+lgy

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### 2 Answers

You need to use logarithmic identities, hence, you need to convert the logarithm of the product in a sum of logarithms, such that:

`lg(x*y) = lg x + lg y`

`lg(x*y) = 9 => lg x + lg y = 9`

You need to solve for x and y the following systemof simultaneous equations, such that:

`{(lg x + lg y = 9),(lg x - lg y = 3):}`

You need to perform the addition of equations, such that:

`lg x + lg y + lg x - lg y = 9 + 3 => 2lg x = 12 => lg x = 6 => x = 10^6 => x = 1 000 000`

Replacing 6 for `lg x` in any of two equations, yields:

`6 + lg y = 9 => lg y = 9 - 6 => lg y = 3 => y = 10^3 => y = 1000`

**Hence, evaluating the solutions x and y yields `x = 10^6` and `y = 10^3` .**

log (xy)=9

`10^9=xy...................(i)`

logx=3+logy

logx-logy=3

log(x/y)=3

10^3=(x/y)=xy^(-1) (ii)

multiply (i) and (ii)

10^12=x^2

taking square roo

10^6=x

substitute in (ii),

we have

10^3=y