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For example with $5^x=125$. How do you manipulate the equation to find $x$?

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    $\begingroup$ In this case you are probably expected to recognize that $5^3=125$. $\endgroup$ Mar 9, 2015 at 5:25

3 Answers 3

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we have: $$5^x = 125$$ $$\log(5^x) = \log(125)$$ $$x\log(5) = \log(125)$$ $$x = \frac{\log(125)}{\log(5)}$$ Can you guess the answer ?

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  • $\begingroup$ oh yeah, thank you. just fixed it $\endgroup$
    – Alexander
    Mar 9, 2015 at 5:30
  • $\begingroup$ Knew the answer, just used a simple example $\endgroup$
    – Ray Kay
    Mar 9, 2015 at 7:34
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By taking logarithm for both sides, we should have: $$\log (5^x) = \log(125)$$

one of the main properties of logarithm functions is that you can take the power out of the logarithm, so we would have:

$$x \log(5) = \log(125)$$

solve for $x$ and you're done..

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Here is one possible solution: $$\begin{aligned}5^x&=125\\ {5^x}/5&={125}/5\\ 5^{x-1}&=25\\ {5^{x-1}}/5&=5\\ 5^{x-2}&=5\\ \end{aligned}$$ We can conclude that $x-2=1 \implies x=3$.

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  • $\begingroup$ One is one of those so-called special cases. :) $\endgroup$
    – Jasha
    Mar 9, 2015 at 5:46
  • $\begingroup$ 125/5 = 25 @Jasha $\endgroup$ Mar 9, 2015 at 5:57
  • $\begingroup$ I've suggested an edit.. @Jasha $\endgroup$ Mar 9, 2015 at 5:59

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