Question 886286
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.