Question 112361
Given:
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{{{7^x-3/4 = sqrt(6)}}}
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Get rid of the {{{-3/4}}} on the left side by adding {{{3/4}}} to both sides to get:
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{{{7^x = 3/4 + sqrt(6)}}}
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The right side of this equation is just a number. We can find that number using a calculator.
First, the fraction {{{3/4}}} has a decimal equivalent of 0.75. (You can find that by using
a calculator to divide 4 (the denominator) into 3 (the numerator) and the calculator 
will give you the answer of 0.75
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Then you can use a calculator to find {{{sqrt(6)}}}. The answer will be 2.449489743.
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So the right side of the equation is 0.75 + 2.449489743 = 3.199489743
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Therefore the equation is now:
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{{{7^x = 3.199489743}}}
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You can solve this equation by taking the log to the base 10 of both sides to make the
equation become:
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{{{log(10,7^x) = log(10,3.199489743)}}}
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On the right side you can use a calculator to find log(3.199489743). This log is 0.505080722.
Substitute this value for the right side and the equation becomes:
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{{{log(10,7^x) = 0.505080722}}}
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When you take the log of a quantity raised to an exponent, you can bring the exponent out
as a multiplier of the log.  This means that on the left side of the equation, the exponent
x can be brought out as the multiplier of the log to give:
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{{{x*log(10,7) = 0.505080722}}}
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Use a calculator to find {{{log(10,7)}}}. If you do you will find that it is 0.84509804.
Substitute this for the log and the equation becomes:
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{{{x*0.84509804 = 0.505080722}}}
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Solve for x by dividing both sides of this equation to get:
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{{{x = 0.505080722/0.84509804 = 0.597659322}}}
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You can check this answer by using a calculator to raise 7 to the exponent 0.597659322 
If you do, you will find that the answer is 3.199489741. Take 3/4 (or 0.75) from this 
and you get that the left side of the original equation is 2.449489741. How does this left
side compare with {{{sqrt(6)}}} which is the right side of the original equation? 
As we found before, {{{sqrt(6) = 2.449489743}}} ... close enough. So the answer of x = 0.597659322
is correct.
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Hope this helps you to understand the problem.
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