4 times 2^x = 8 times 5^x solve for x

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- Aug 29th 2009, 03:12 PMlbj2322logs!!
4 times 2^x = 8 times 5^x solve for x

- Aug 29th 2009, 03:57 PMQM deFuturo
$\displaystyle 4(2^x) = 8(5^x) $

Since both sides are equal, if we take the log of both sides, they remain equal. Assume the common (base 10) log

$\displaystyle log (4(2^x)) = log (8(5^x)) $

Hopefully you know the properties of logs, and know how to re-write a log of a product:

$\displaystyle log 4 + log 2^x = log 8 + log 5^x $

Another property of logs lets you move the exponent

$\displaystyle log 4 + x log 2 = log 8 + x log 5 $

We've now cleared the exponents. You should be able to move all the x's to the same side, and the other terms to the other side, and use basic algebra to get x by itself. Then use your calculator to compute the $\displaystyle log_{10} $ values.

Note also, you can simplify the original equation by first dividing both sides by 4.