My son gave me this question to solve for r:
3[sup]3log[sub]9[/sub]4[/sup] = r[sup]1/2[/sup]
It was given to him by a friend in university.
Solving for r is straightforward. I have tried to calculate the actual value for r by pen and paper without success. Using a calculator gives the answer, r = 64.
Is there a way to simplify this down without a calculator?
And, why yes, I am quite proud of my subscript/superscript coding.
3[sup]3log[sub]9[/sub]4[/sup]
= 3[sup]log[sub]9[/sub]64[/sup]
= 3[sup]log[sub]3[/sub]64/log[sub]3[/sub]9[/sup]
= 3[sup]log[sub]3[/sub]64/log[sub]3[/sub]9[/sup]
= 3[sup]log[sub]3[/sub]64/2[/sup]
= 3[sup]log[sub]3[/sub]64[sup]1/2[/sup][/sup]
= 64[sup]1/2[/sup]
= r[sup]1/2[/sup] iff r = 64
Edited to add: I guess the important step is the change of base formula on the third line.
Thank you!
leahcim gives the correct answer, and what I say will only be a re-framing of this:
Since the right-hand side is the square root of what we are seeking, we may as well begin by squaring both sides. We can conceptualize this as each factor of 3 on the left-hand side becoming a factor of 3[sup]2[/sup] (i.e., 9) instead. Thus, we get:
9[sup]3log[sub]9[/sub]4[/sup] = r.
At this point, you likely already know how to deal with the left-hand side: 9[sup]log[sub]9[/sub]4[/sup] is, by definition, 4, and since we actually have 3 times this many factors of 9 on our left-hand side, we end up with the product of 3 many 4s:
4[sup]3[/sup] = r.