Can $5 x^{4} = 3^{x}$ $5x^4=3^x$ be solved algebraically?

Question

I solved a problem similar to this using graphing but are there algebra techniques for solving an equation of this form?

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Answer 1 · 2017-01-10T02:21:40

With powers, the remedy is often logaritms.

Use:
$log (A \cdot B) = log A + log B$ $log(A*B)=logA+logB$ and ${log A}^{B} = B log A$ $logA^B=BlogA$

$log (5 x^{4}) = log (3^{x}) \to$ $log(5x^4)=log(3^x)->$
$log 5 + 4 log x = x log 3 \to$ $log5+4logx=xlog3->$

$log 5 = x log 3 - 4 log x$ $color(red)(log5=xlog3-4logx)$

$log 5 = {log 3}^{x} - {log x}^{4}$ $color(red)(log5=log3^x-logx^4)$

$log 5 = log (\frac{3^{x}}{x^{4}})$ $color(red)(log5=log(3^x/x^4)$

and now I'm just going in circles