logarithm question A level maths

Question

The question is : Given that log(x^2y) = 4 and log(x^4/y^3) = 18, find the values of log(x) and log(y).

The answers are log(x) = 3 and log(y) = -2 but I just can't seem to get those answers whatsoever, can someone pls kindly help?

Answer 1

First, discuss what the expression x^2y means. Is it x^2y or x²y?

Fortunately, we know the answers, and we can check them.

log(x) = 3, so x = 1000; log(y) = -2, so y = 0.01.

Then log(x^2y) = log(1000^0.02) = log(10^0.06) = 0.06, but log(x²y) = log(1000² * 0.01) = log(10⁴) = 4.

Similar, x^4/y^3 should mean x⁴/y³,

then log(x⁴/y³) = log(1000⁴/0.01³) = log(10¹²/10^-6) = log(10¹⁸) = 18.

Now we have found the correct interpretation and could solve the problem.

log(x²y) = 4 and log(x⁴/y³) = 18.

Use properties of logarithms.

log(x²y) = 2log(x) + log(y), log(x⁴/y³) = 4log(x) – 3log(y).

We have 2 equations:

2log(x) +   log(y) = 4

4log(x) – 3log(y) = 18

If we use the substitution a = log(x), b = log(y), we obtain the linear system in two variables:

2a +   b =  4

4a – 3b = 18

I am sure that you are able to solve linear systems. The solution is a = 3, b = -2

Finally, log(x) = 3, log (y) = -2.