How do I solve this log equation?

Question

How do I solve this log equation? I think it's impossible to solve this. Correct me of I'm wrong.

$$ 8\log_{10}\left(\dfrac {50-t}{45-t}\right)=5\log_{10}\left(\dfrac {50-t}{40-t}\right)$$

Answer 1

You can move the coefficients into the log to get $$\log \left(\frac{50-t}{45-t}\right)^8 = \log \left(\frac{50-t}{40-t}\right)^5.$$ (Note that G. Sassatelli in the comment below is right that this might introduce extra spurious solutions, since solutions to the new equation where $\frac{50-t}{45-t}$ is negative are not solutions of the original equation. Therefore any solution to the new equation with $45\leq t \leq 50$ should be discarded at the end of the calculation.)

You can then exponentiate both sides, eliminate the common numerator, and rearrange to get the polynomial equation $$(40-t)^5(50-t)^3 - (45-t)^8 = 0.$$ Things don't look too promising from here, but you can use e.g. Wolfram Alpha to approximate a solution $t\approx 55.431.$