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January 26th, 2016, 09:42 AM  #1 
Senior Member Joined: Jan 2012 Posts: 109 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: 2,761 Thanks: 1416 
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: 109 Thanks: 2  Thanks. The problem is now solved.


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