2010 AMC 12B Problems/Problem 12
Problem 12
For what value of 
does
![\[\log_{\sqrt{2}}\sqrt{x}+\log_{2}{x}+\log_{4}{x^2}+\log_{8}{x^3}+\log_{16}{x^4}=40?\]](http://latex.artofproblemsolving.com/e/0/c/e0c9ec94030e09b44a3086de323084adbecfb713.png)
Solution
Using logarithm rules: \begin {align*} \log_{\sqrt{2}}\sqrt{x} + \log_2x + \log_4(x^2) + \log_8(x^3) + \log_{16}(x^4) & = 40 \\ \frac{1}{2} \frac{\log_2x}{\log_2\sqrt{2}} + \log_2x + \frac{2\log_2x}{\log_24} + \frac{3\log_2x}{\log_28} + \frac{4\log_2x}{\log_216} & = 40 \\ \log_2x + \log_2x + \log_2x + \log_2x + \log_2x & = 40 \\ 5\log_2x & = 40 \\ \log_2x & = 8 \\ x & = 256 (D) \end{align*}
See also
| 2010 AMC 12B (Problems • Answer Key • Resources) | |
| Preceded by Problem 12 |
Followed by Problem 14 |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
| All AMC 12 Problems and Solutions | |
