Question

How do you solve for x and y: `log(x^2y^3)=7` and `log(x/y)=1`?

Answer 1

`x=100 and y=10`. The graph plots the solution point (100, 10)

Explanation:

For the logarithms to be real, both x and y > 0.

Algebraic method:

Expanding in terms of log x and log y,

`2 log x + 3 log y = 7 and`

#log x - log y =1. Solving for log x and log y,

log x =2 and log y = 1. Inverting,

`x = 10^2 =100 and y = 10^1=10`.

Graphical method:

The first equation has the non-logarithmic form

`y = 10^(7/3) x^(-2/3) in Q_1`

The second reduces to `y = x/10 in Q_1`.

Look for common point at (100, 10), in the combined graph.
graph{(y-10^(7/3)x^(-2/3))(x-10y)((x-100)^2+(y-10)^2-.01)=0[90 110 5 15]}