Question

The least value of ${\log _2}(x) - {\log _x}(0.125)$ for $x > 1$ is equal to

Answer 1

We have a logarithmic function in base two and another logarithmic function in base x. We have to use the concept of AM and GM in this question to solve it. Without it, we can’t solve it easily.

Complete step by step solution:
According to the question we have to find the least value of ${\log _2}(x) - {\log _x}(0.125)$ for $x > 1$
So we will use the concept that arithmetic mean is greater than geometric mean, that is $AM \ge GM$
Now we have to simplify the equation to use this concept
$\Rightarrow {\log _2}(x) + {\log _x}(\dfrac{{1000}}{{125}})$ (Convert 0.125 into fraction and inverse it and make the negative sign to positive)
$\Rightarrow {\log _2}(x) + {\log _x}(8)$ (1000 divided by 125 will give 8)
$\Rightarrow {\log _2}(x) + {\log _x}({2^3})$ (8 is nothing but two raised to three)
$\Rightarrow {\log _2}(x) + 3{\log _x}(2)$ (A number in power can be taken down as multiplication with the log)
Now, here we can apply the concept that $AM \ge GM$
Here AM is arithmetic mean = $\dfrac{{{{\log }_2}(x) + 3{{\log }_x}(2)}}{2}$
And, GM is geometric mean = $\sqrt {{{\log }_2}(x) \times 3{{\log }_x}(2)} = \sqrt 3$
And $AM \ge GM$ so
$\Rightarrow \dfrac{{{{\log }_2}(x) + 3{{\log }_x}(2)}}{2} \ge \sqrt 3$
$\Rightarrow {\log _2}(x) + 3{\log _x}(2) \ge 2\sqrt 3$

Hence, the least value of the equation is $2\sqrt 3$
It is our answer.

Note:
We have used the terms such as arithmetic mean which is the average of a set of numerical values, as calculated by adding them together and dividing by the number of terms in the set. And geometric mean, which is a type of average, usually used for growth rates, like population growth or interest rates. While the arithmetic mean adds items, the geometric mean multiplies items. Also, you can only get the geometric mean for positive numbers. We can also observe the log with base of two and other with base of x.