While reading through my textbook, I came across this particular equation: $$ x = x\log (y) + \log (y) $$ But they solve it by doing this: $$ x = x\log (y) + \log (y) $$ $$ x = (x + 1)\log(y) $$ $$ \frac {x}{x+1} = \log(y) $$ $$ y = 10^{\frac{x}{x+1}} $$ Which is fine, but I don't understand why they didn't do it like this: $$ x = x\log (y) + \log (y) $$ $$ x=\log(y^x)+\log(y) $$ $$ x=\log(y^{2x}) $$ $$ x=2x\log(y) $$ $$ \frac{x}{2x}=\log(y) $$ $$ y=10^{\frac{x}{2x}} $$ I'm confused over which one is correct.
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$\begingroup$ Yep, I feel like an idiot! Thanks, shost71 and MPW! $\endgroup$– dr_nefarioMar 16, 2015 at 2:34
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3 Answers
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I think you have got an error: $ x = log(y^x) + log(y) = log(y^x\cdot y) = log(y^{x+1})$
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You've made an error.
$\log y^x +\log y$ is $\log y^{x+1}$, not $\log y^{2x}$
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When you do $$ log(y^x) + log(y) $$ this equals: $$ log(y^x \cdot y) = log(y^{x+1}) $$
which gives $ y = 10^{\frac{x}{x+1}} $
