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January 26th, 2016, 09:42 AM  #1 
Senior Member Joined: Jan 2012 Posts: 140 Thanks: 2  A queston on logarithms
Hello, Can one help me solve one logarithms' question: Given aand b are positive numbers satisfying $\displaystyle 4(\log_{10}a)^{2}+(\log_{2}b)^{2}=1$, then which of the following statement(s) is/are correct? (A) Greatest and least possible values of 'a' are reciprocal of each other. (B) Greatest and least possible values of 'b' are reciprocal of each other. (C) Greatest value of 'a' is the square of the largest value of 'b'. (D) Least value of 'b' is the square of the least value of 'a'. Thanks. 
January 26th, 2016, 10:57 AM  #2 
Math Team Joined: Jul 2011 From: Texas Posts: 3,002 Thanks: 1587 
let $x=\log_{10}{a}$ $y = \log_2{b}$ $4x^2+y^2 = 1$, an ellipse such that $\dfrac{1}{2} \le x \le \dfrac{1}{2}$ and $1 \le y \le 1$ $\dfrac{1}{2} \le \log_{10}{a} \le \dfrac{1}{2}$ $1 \le \log_2{b} \le 1$ ... enough information for you to continue? 
January 27th, 2016, 05:06 AM  #3 
Senior Member Joined: Jan 2012 Posts: 140 Thanks: 2  Thanks. The problem is now solved.


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