AMC12 2010 B
AMC12 2010 B · Q12
AMC12 2010 B · Q12. It mainly tests Logarithms (rare), Manipulating equations.
For what value of $x$ does
\[\log_{\sqrt{2}}\sqrt{x}+\log_{2}{x}+\log_{4}{x^2}+\log_{8}{x^3}+\log_{16}{x^4}=40?\]
对于什么值的 $x$，有
\[\log_{\sqrt{2}}\sqrt{x}+\log_{2}{x}+\log_{4}{x^2}+\log_{8}{x^3}+\log_{16}{x^4}=40?\]
(A)
8
8
(B)
16
16
(C)
32
32
(D)
256
256
(E)
1024
1024
Answer
Correct choice: (D)
正确答案：（D）
Solution
\[\log_{\sqrt{2}}\sqrt{x} + \log_2x + \log_4(x^2) + \log_8(x^3) + \log_{16}(x^4) = 40\]
\[\frac{1}{2} \frac{\log_2x}{\log_2\sqrt{2}} + \log_2x + \frac{2\log_2x}{\log_24} + \frac{3\log_2x}{\log_28} + \frac{4\log_2x}{\log_216} = 40\]
\[\log_2x + \log_2x + \log_2x + \log_2x + \log_2x = 40\]
\[5\log_2x = 40\]
\[\log_2x = 8\]
\[x = 256 \;\; (D)\]
\[\log_{\sqrt{2}}\sqrt{x} + \log_2x + \log_4(x^2) + \log_8(x^3) + \log_{16}(x^4) = 40\]
\[\frac{1}{2} \frac{\log_2x}{\log_2\sqrt{2}} + \log_2x + \frac{2\log_2x}{\log_24} + \frac{3\log_2x}{\log_28} + \frac{4\log_2x}{\log_216} = 40\]
\[\log_2x + \log_2x + \log_2x + \log_2x + \log_2x = 40\]
\[5\log_2x = 40\]
\[\log_2x = 8\]
\[x = 256 \;\; (D)\]
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