Question 1209181
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Answer: <font color=red size=4>1</font>


Explanation


Let's isolate x so we can determine 1/x.
{{{2^x = 12}}}


{{{log((2^x)) = log((12))}}} Apply logs to both sides.


{{{x*log((2)) = log((12))}}} Use the rule log(M^N) = N*log(M) to pull down the exponent.


{{{x = log((12))/log((2))}}} Divide both sides by log(2).


{{{1/x = log((2))/log((12))}}} Apply the reciprocal to both sides.



{{{6^y = 12}}} can be rearranged into {{{1/y = log((6))/log((12))}}} when following similar steps.


Then,
{{{1/x + 1/y = log((2))/log((12)) + log((6))/log((12))}}}


{{{1/x + 1/y = ( log((2))  + log((6)) )/log((12))}}}


{{{1/x + 1/y = ( log((2*6)) )/log((12))}}} Use the rule log(M)+log(N) = log(M*N) where the base of each log must be the same. 


{{{1/x + 1/y = log((12))/log((12))}}}


{{{1/x + 1/y = 1}}}


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Another approach.


2^x = 12 rearranges to 2 = 12^(1/x) when raising both sides to the 1/x power. 
Similarly we can say that 6^y = 12 turns into 6 = 12^(1/y)


We have these new equations
2 = 12^(1/x)
6 = 12^(1/y)


These two new equations involve exponents 1/x and 1/y. 
If we multiply straight down then we'll be able to add these exponents due to the rule a^b*a^c = a^(b+c).


The left hand sides multiply to 12 aka 12^1.
The right hand sides multiply to 12^( (1/x) + (1/y) )
We form the equation 12^1 = 12^( (1/x) + (1/y) )


Comparing the two sides shows that the bases are both 12, so the exponents must be equal.


Therefore (1/x) + (1/y) = <font color=red>1</font>


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Let's confirm the answer. 


Use a calculator and the change of base formula to determine these two approximations
x = log<sub>2</sub>(12) = log(12)/log(2) = 3.584963
y = log<sub>6</sub>(12) = log(12)/log(6) = 1.386853
Each approximate value is rounded to 6 decimal places.


Then,
(1/x) + (1/y) = (1/3.584963) + (1/1.386853)
(1/x) + (1/y) = 0.999999860928
We don't land on 1 exactly but we get close enough. 
The rounding error will depend how you rounded the approximations for x and y. 


If you were to use more decimal digits in the values of x and y, then you'll get closer to 1.
Let's say we rounded to 12 decimal places (rather than 6)
(1/x) + (1/y) = (1/3.584962500721) + (1/1.386852807235)
(1/x) + (1/y) = 0.9999999999997738 
We get much closer to 1 this time. 
There are 12 copies of "9" listed in that number shown above.
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