Solve for x and y: 10^(2x-y)=1000 and logx - logy=log4

Question

10^(2x-y) = 1000 and log x - log y = log 4, find x and y.

Accepted Answer

x = 8, y = 2

Answer

Step 1: Simplify the exponential equation. Since 1000 = 10^3, set the exponents equal: 2x - y = 3. Step 2: Simplify the logarithmic equation using the quotient rule of logarithms: log(x/y) = log4. Since log functions are one-to-one, x/y = 4, so x = 4y. Step 3: Substitute x = 4y into 2x - y = 3: 2(4y) - y = 3 → 8y - y = 3 → 7y = 3? Correction: 2(4y)-y=8y-y=7y=3 is wrong, 10^3=1000 so 2x-y=3, x=4y, substitute: 2*(4y)-y=8y-y=7y=3? No, 10^(2x-y)=10^3 so 2x-y=3, x=4y, 2*(4y)-y=8y-y=7y=3 → y=3/7? No, mistake: logx - logy=log4 → log(x/y)=log4 → x/y=4 → x=4y. 10^(2x-y)=1000=10^3 → 2x-y=3. Substitute x=4y: 2*(4y)-y=8y-y=7y=3 → y=3/7, x=12/7. Wait no, 10^(2x-y)=1000 → 2x-y=3, x=4y, 2*(4y)-y=8y-y=7y=3 → y=3/7, x=12/7. But let's check: 10^(2*(12/7)-3/7)=10^(21/7)=10^3=1000, correct. log(12/7)-log(3/7)=log((12/7)/(3/7))=log4, correct. So correct answer x=12/7, y=3/7. Step 4: Verify the solutions in both original equations to confirm validity.