# How do I solve this log equation? [closed]

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

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

## closed as off-topic by Namaste, Davide Giraudo, zoli, Henrik, C. FalconApr 2 '17 at 23:16

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• The Base is of log is 10 – Aditya DS Apr 2 '17 at 19:28
• @projectilemotion It's irrelevant. – user228113 Apr 2 '17 at 19:29
• @G.Sassatelli why? – Aditya DS Apr 2 '17 at 19:30
• How can I solve this? – Aditya DS Apr 2 '17 at 19:30
• @AdityaDS The logarithm in one base is a constant times the logarithm in an other base. So if you change the base of the logarithm on both sides, you multiply both sides by the same number. – Paul Apr 2 '17 at 19:37

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 $t\approx 55.431.$
• If we are to solve it numerically at the end, then the initial equation is numerically more stable compared to powers of $8$ in this one. This appear to be progress, but ironically having the $8$ and $5$ outside the log help for a better convergence. Sometimes manual solving and numerical solving are antagonist. :p – zwim Apr 2 '17 at 21:56