Question

How do you solve a system of logarithmic equations? They have completely different bases:

{(log(25)^3)x + (log(2)^7)y = log(5)^27
{(log(7)^8)x + (log(3)^5)y = log(49)^2

Answer 1

recall that log_a(b) = ln(a)/ln(b)

So, converting everything to natural logs (or any set of logs with a common base),

ln3/ln25 x + ln7/ln2 y = ln27/ln5
ln8/ln7 x + ln5/ln3 y = ln2/ln49

Now just solve the system as usual. It's messy, but it's just numbers. If you don't need an exact solution, just convert everything to decimal coefficients.