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Without using logarithms, how can I solve

$$10=\left(\frac{6}{5}\right)^x$$

Raffaele
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Peppers
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    Why should we not use the obvious approach to solve this ? An alternative could be Newton's method as a numerical approach. – Peter Nov 19 '20 at 16:08
  • $(6/5)^4=1296/625\gt1250/625=2$,so $(6/5)^{16}\gt2^4\gt10$, so $x\lt16$ – Empy2 Nov 19 '20 at 16:54
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    Since the answer involves logarithm, I don't see how an analytical approach could avoid it. Your only option is numerical solution. – Andrei Nov 19 '20 at 16:57
  • Trial and error for 58 minutes $x\approx 12.6293$ – Raffaele Nov 19 '20 at 17:17

1 Answers1

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You could use the series expansion of the natural logarithm as follows:

$$ \frac{\sum\limits_{n=1}^{\infty}\frac{1}{n}\left(1-\frac{1}{10}\right)^{n}}{\sum\limits_{n=1}^{\infty}\frac{1}{n}\left(1-\frac{5}{6}\right)^{n}} $$

Therefore, you don't use the logarithm directlry. But I am not sure, if this infinite sum is better for you. At least it will approach the correct result, the more terms you use.

thinkingeye
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