SOLUTION: 2^(log_2(3)-log_8(9))= so far i used the change of base to get 2^(ln3/ln8)= the answer is 3root3 ->^3√3 but i don't know how to get that from here.

Question 886286: 2^(log_2(3)-log_8(9))=
so far i used the change of base to get
2^(ln3/ln8)=
the answer is 3root3 ->^3√3 but i don't know how to get that from here.
Found 2 solutions by rothauserc, Theo:
Answer by rothauserc(4718) (Show Source): You can put this solution on YOUR website!
note that the logarithm quotient rule requires both logs to have the same base
2^(log_2(3)-log_8(9)) = 2^( 1.5849625007 - 1.0566416671) =
2^(0.528320834) = 1.44224957

Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
i didn't get 3 * sqrt(3) as the value for 2^(log2(3) - log8(9))
my first look was to use the calculator and convert the logs to base 10 which the calculator can handle.

i got:
log2(3) = log10(3)/log10(2) = 1.584962501
log8(9) = log10(9)/log10(8) = 1.056641667

putting these into the original equation and i got:
2^(log2(3)-log8(9)) = 2^(1.584962501 - 1.056641667) = 2^(.5283208336) which is equal to 1.44224957

so your solution should be 1.44224957

now to solve it the long way that takes a lot more work but may also be instructive.

log8(9) = y if and only if 8^y = 9
since 8 is equal to 2^3, this equation becomes:
(2^3)^y = 9 which is equivalent to 2^3y = 9
2^3y = 9 if and only if log2(9) = 3y
divide both sides of this equation by 3 to get y = log2(9)/3

you have both log8(9) and log2(9)/3 equal to y so these 2 expressions are also equal to each other.

replace log8(9) by log2(9)/3 in your original equation and you get:

2^(log2(3) - log2(9)/3)

place log2(3) - log2(9)/3 under the same denominator and you get:
2^((3*log2(3) - log2(9)) / 3)

since a*log(b) = log(b^a), you can simplify this to get:
2^(log2(3^3) - log2(9)) / 3)

simplify this further to get:
2^((log2(27) - log2(9)) / 3)

since log(a) - log(b) = log(a/b), this expression becomes:
2^(log2(27/9)/3)

simplify this further to get:
2^(log2(3)/3)

once you get to this point, you still have to do some conversions.
suffice it to say that 2^(log2(3)/3) = 1.44224957, same as we got originally.

if you use your calculator and take the cube root of 3, you will see that cube root of 3 is equal to 1.44224957.

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SOLUTION: 2^(log_2(3)-log_8(9))= so far i used the change of base to get 2^(ln3/ln8)= the answer is 3root3 ->^3&#8730;3 but i don't know how to get that from here.

SOLUTION: 2^(log_2(3)-log_8(9))= so far i used the change of base to get 2^(ln3/ln8)= the answer is 3root3 ->^3√3 but i don't know how to get that from here.