Question 124443
Here's a way that it can be done.
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Given:
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{{{y =log(6,3.5) + log(6,2)}}}
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By the rules of logarithms, if you have the sum of two logs (must be the same base) you
can apply the product rule:
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{{{y = log(b,A) + log(b,C) = log(b,(A*C))}}}
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By comparing this rule term by term with the given problem, you can see that b = 6, A = 3.5, and C = 2
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Substituting these values into the rule, the problem rule gives:
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{{{y = log(6,3.5) + log(6,2) = log(6,(3.5*2))}}}
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and when you multiply out the two terms you get:
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{{{y = log(6,7)}}}
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Next you can convert this logarithmic for to exponential form by applying the conversion rule:
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{{{y = log(b,D)}}} is equivalent to the exponential form {{{b^y = D)}}}
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If you compare this form to the logarithmic form you now have you can see that y = y, b = 6, and D = 7
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Substituting these values into the exponential form you convert your logarithmic form into the
exponential form:
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{{{6^y = 7}}}
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Now take the log to the base 10 of both sides of this exponential form and you get:
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{{{log(10,6^y) = log(10,7)}}}
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Applying the exponential rule to the left side brings the exponent out as a multiplier and
the equation then becomes:
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{{{y*log(10,6) = log(10,7)}}}
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You can now use a calculator to find {{{log(10,6)}}}. Enter 6 and press the "log" key. You should
find that the value you get for this log is 0.77815125.
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Similarly you can find {{{log(10,7)}}} by entering 7 and pressing the "log" key to get that
this log is 0.84509804.
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Substitute the two values you have for the base 10 logs of 6 and 7 into the equation and
the equation becomes:
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{{{y*(0.77815125) = 0.84509804}}}
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Solve for y by dividing both sides by 0.77815125 and you get:
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{{{y = 0.84509804/0.77815125 = 1.086033133}}}
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Since way back at the beginning of the problem we defined y as {{{log(6,3.5) + log(6,2)}}}
you can now say that:

 {{{log(6,3.5) + log(6,2) = 1.086033133}}} 
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and this rounds off to:
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{{{log(6,3.5) + log(6,2) = 1.09}}}
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The answer is answer A ... 1.09
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Hope this helps you to find where you went astray in working the problem.
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