Question 86947
Anytime you see an equation that has a variable in the exponent, one of the first things
you should think of is "logarithms"
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You are given:
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{{{(125)^x=625}}}
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Take the logarithm of both sides of this equation. You can use base 10 or base e, which is 
"ln" or natural logarithms. These can be worked on a normal scientific calculator so either
is a good choice. Let's choose base 10. Taking the Log base 10 of both sides of the equation
results in:
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{{{Log[10]125^x = Log[10]625}}}
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To simplify things a little, let's use a calculator to find {{{Log[10]625}}}. Enter 625
on the calculator and press the "log" key. You should get 2.795880017. Substituting
this value results in the equation becoming:
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{{{Log[10]125^x = 2.795880017}}}
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Next we'll use another property of logarithms. If you are finding the logarithm of a quantity
that has an exponent, an equivalent form is to multiply the logarithm by the exponent.
In other words, {{{Log[10]125^x}}} is equivalent to {{{x*Log[10]125}}} so let's substitute
that into our equation to get:
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{{{x*Log[10]125 = 2.795880017}}}
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Now use a calculator to find {{{Log[10]125}}}. Enter 125 and press the "log" key to get
{{{Log[10]125 = 2.096910013}}}. Substitute this into the equation and you now have:
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{{{x*2.096910013 = 2.795880017}}}
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Finally divide both sides by 2.096910013 and the answer becomes:
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{{{x = 2.795880017/2.096910013 = 1.333333333}}} and this is {{{4/3}}}
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Hope this helps you with your understanding of logarithms.