Question 115400
The first of these problems uses the following definition that relates the logarithmic form 
to the exponential form:
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The logarithmic form {{{log(a,B)= y}}} is equivalent to the exponential form {{{a^y = B}}}
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Let's use this relationship on the problem. You are given:
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{{{log(10,x = 3)}}}
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Compare this to the logarithmic form above. If you do you will see that "a" in the logarithmic
form is 10 in your problem. Also you will see that y in the logarithmic form is 3 in your
problem. And B in the logarithmic form is x in your problem. Now all you have to do to convert
your problem is to go to the exponential form and plug in the corresponding values.
Start with the exponential form {{{a^y = B}}} and substitute 10 for "a", 3 for y, and x for B.
When you do those substitutions you get as the exponential form of the given problem:
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{{{10^3 = x}}}
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Cubing 10 results in 1000 and the exponential form reduces to just:
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{{{1000 = x}}}
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That is the answer to the first problem ... x = 1000. This is answer c in your list of
answers.
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The second problem first involves taking the logarithm of both sides to get:
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{{{log(10,9^x) = log(10,11)}}}
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On the left side you can use the logarithmic property that the log of a quantity that is raised 
to an exponent is equal to the logarithm of the quantity multiplied by the exponent.
This means that {{{log(10,9^x) = x*log(10,9) }}}. Substitute this into the equation and
you then have:
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{{{x*log(10,9) = log(10,11)}}}
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Now you can use a calculator to find that:
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{{{log(10,9) = 0.954242509}}} and 
{{{log(10,11) = 1.041392685}}}
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Substitute these values into the equation and you have:
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{{{x*0.954242509 = 1.041392685}}}
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Solve for x by dividing both sides by 0.954242509 to get:
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{{{x = 1.041392685/0.954242509}}}
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After you do the division on the right side ... using a calculator ... you have:
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{{{x = 1.09132917}}}
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Answer b in your list of answers is the correct choice.
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Hope this helps you to understand logarithms a little bit more.
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