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I've been asked to solve for $x\,$ in

$5^x + 4·5^{x+1} = 63$

The answer is $x = \frac{\log3}{\log5}$

I cannot do this without a calculator. Is there a particular method I should be using to approach this? The calculator simplifies the problem to

$21·5^x = 63$

From here it is obvious how to solve the problem. I just don't understand how I could get to that point without a calculator. Any help would be appreciated. Thanks!

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    $\begingroup$ Hint: $5^{x+1}=5\cdot 5^x$ $\endgroup$ – lattice Jun 7 '16 at 13:48
  • $\begingroup$ ...that makes it so obvious. Thank you! $\endgroup$ – Peter Jun 7 '16 at 14:03
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You use the properties of logarithms. Here you need to recognize $$4(5^{x+1})=4\cdot 5 \cdot 5^x$$, then use the distributive property on $5^x$ to get there.

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$5^{x+ 1}= 5^x(5^1)= 5(5^x)$ so $5^x+ 4(5^{x+ 1})= 5^x+ 4(5)(5^x)= 5^x+ 20(5^x)= 21(5^x)= 63$

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