Question 253720
If {{{a=log(7,(11-6*sqrt(2)))}}} and {{{b=log(7,(45+29*sqrt(2)))}}}, find 3a+2b in the simplest form.


{{{a = log(7,(11-6*sqrt(2)))}}} if and only if {{{7^a = 11-6*sqrt(2)}}}


{{{b = log(7,(45+29*sqrt(2)))}}} if and only if {{{7^b = 45+29*sqrt(2)}}}


take log of both sides of first equation to get:


{{{7^a = 11-6*sqrt(2)}}} becomes:


{{{log((7^a)) = log((11-6*sqrt(2)))}}}


this becomes:


{{{a*log((7)) = log((11-6*sqrt(2)))}}}


divide both sides by log((7)) to get:


{{{a = log((11-6*sqrt(2)))/log((7))}}} *************


take log of both sides of second equation to get:


{{{7^b = 45+29*sqrt(2)}}} becomes:


{{{log((7^b)) = log((45 + 29*sqrt(2)))}}}


this becomes:


{{{b*log((7)) = log((45 + 29*sqrt(2)))}}}


divide both sides by log((7)) to get:


{{{b = log((45+29*sqrt(2)))/log((7))}}}


{{{3a + 2b = 3*log((11-6*sqrt(2)))/log((7)) + 2*log((45+29*sqrt(2)))/log((7))}}}


since the denominator is the same, we get:


{{{3a + 2b = (3*log((11-6*sqrt(2))) + 2*log((45+29*sqrt(2))))/log((7))}}}


I don't think you can simplify it any further, but you can solve it using the LOG function of your calculator.


equation becomes:


{{{3a + 2b = (3*.400489398 + 2*1.934560022)/.84509804}}}


this becomes:


{{{3a + 2b = (1.201468195 + 3.869120045)/.84509804 = 6}}}


I confirmed the answer is correct by using the logarithm base conversion formula to convert base 7 logarithm to a base 10 logarithm.


that conversion formula is:


{{{log(7,(x)) = log(10,(x))/log(10,(7))}}}


you could also have solved this problem directly by using the base conversion formula.