2010 AMC 12B Problems/Problem 12: Difference between revisions
Created page with '== Problem 12 == For what value of <math>x</math> does <cmath>\log_{\sqrt{2}}\sqrt{x}+\log_{2}{x}+\log_{4}{x^2}+\log_{8}{x^3}+\log_{16}{x^4}=40?</cmath> <math>\textbf{(A)}\ 8 \…' |
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== Solution == | == Solution == | ||
<cmath> \log_{\sqrt{2}}\sqrt{x} + \log_2x + \log_4(x^2) + \log_8(x^3) + \log_{16}(x^4) = 40 </cmath> | |||
<cmath> \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 </cmath> | |||
\log_{\sqrt{2}}\sqrt{x} + \log_2x + \log_4(x^2) + \log_8(x^3) + \log_{16}(x^4) | <cmath> \log_2x + \log_2x + \log_2x + \log_2x + \log_2x = 40 </cmath> | ||
\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} | <cmath> 5\log_2x = 40 </cmath> | ||
\log_2x + \log_2x + \log_2x + \log_2x + \log_2x | <cmath> \log_2x = 8 </cmath> | ||
5\log_2x | <cmath> x = 256 (D) </cmath> | ||
\log_2x | |||
x | |||
== See also == | == See also == | ||
{{AMC12 box|year=2010|num-b=12|num-a=14|ab=B}} | {{AMC12 box|year=2010|num-b=12|num-a=14|ab=B}} | ||
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)
![\[\log_{\sqrt{2}}\sqrt{x} + \log_2x + \log_4(x^2) + \log_8(x^3) + \log_{16}(x^4) = 40\]](http://latex.artofproblemsolving.com/7/4/9/7498c0fe81193b08592296dd3f814d2cbcdcf655.png)
![\[\frac{1}{2} \frac{\log_2x}{\log_2\sqrt{2}} + \log_2x + \frac{2\log_2x}{\log_24} + \frac{3\log_2x}{\log_28} + \frac{4\log_2x}{\log_216} = 40\]](http://latex.artofproblemsolving.com/f/7/f/f7fcc47e14a25755d1fdb4ab16fad12a4220b1f0.png)
![\[\log_2x + \log_2x + \log_2x + \log_2x + \log_2x = 40\]](http://latex.artofproblemsolving.com/2/4/b/24b4555144e83c662e5e0d838d6e4f66a883de4f.png)
![\[5\log_2x = 40\]](http://latex.artofproblemsolving.com/a/0/f/a0fa15d3c8038c8eeef4f27d4017db1935f5e66b.png)
![\[\log_2x = 8\]](http://latex.artofproblemsolving.com/e/6/9/e69d0058e5864841add8ab316c7bd9daf5355459.png)
![\[x = 256 (D)\]](http://latex.artofproblemsolving.com/6/b/9/6b98248cf47d0ff0f55071f2c555fcb63c52e447.png)
