# Logarithmic Equations

How does one go about solving:

$(5x+2)^{\frac{4}{3}} = 16$

I'm confused as how to parse through the equation to solve it using logs.

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take the 3/4 power in both sides $(5x+2)= 16^{3/4}=8$ (use calculator) :) then $5x=8-2$ and $x=6/5$ –  Jose Garcia May 27 '13 at 18:18
You don't use logs; use logs only when you have a variable in the exponent, as in $2^{2x+5}=100$ or something like that (you really don't need a calculator either for this problem ;) –  Zen May 27 '13 at 18:19
Every comment and solution assumed the power was $\frac 43$ so I edited to make it so. If this is wrong, please let me know what it should be. –  Mark Bennet May 27 '13 at 18:29

\begin{align} &(5x+2)^{\frac{4}{3}} = 16\\\implies &5x+2 = (16)^{\frac{3}{4}}\\\implies &5x+2 = (2^4)^{\frac{3}{4}}\\\implies &5x+2 = 2^3\\\implies &5x+2 = 8\\\implies &5x= 6\\\implies &x=\frac{6}{5} \end{align}

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$$(5x+2)^{\frac{4}{3}} = 16$$ $$\left((5x+2)^{\frac{4}{3}}\right)^{\frac{3}{4}} = 16^{\frac{3}{4}}$$ $$(5x+2)^{\frac{4}{3}\cdot \frac{3}{4}}=2^{4\cdot \frac{3}{4}}$$ $$5x+2=2^3$$ $$5x+2=8$$ $$x=\frac{6}{5}$$

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