Suppose log _a b + log _b a = c. The smallest | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(C) Given $\log _a b + \log _b a = c$.Since $a, b > 1$,both $\log _a b$ and $\log _b a$ are positive real numbers.By the Arithmetic Mean-Geometric Mea

Vedclass · Accepted Answer

$2$

Vedclass · Answer

(C) Given $\log _a b + \log _b a = c$.Since $a, b > 1$,both $\log _a b$ and $\log _b a$ are positive real numbers.By the Arithmetic Mean-Geometric Mean $(AM-GM)$ inequality,for any two positive real numbers $x$ and $y$,$\frac{x+y}{2} \geq \sqrt{xy}$.Let $x = \log _a b$ and $y = \log _b a$.Then $\frac{\log _a b + \log _b a}{2} \geq \sqrt{\log _a b \cdot \log _b a}$.Since $\log _a b = \frac{1}{\log _b a}$,we have $\log _a b \cdot \log _b a = 1$.Thus,$\frac{c}{2} \geq \sqrt{1} = 1$.This implies $c \geq 2$.The smallest possible integer value of $c$ is $2$.

Suppose $\log _a b + \log _b a = c$. The smallest possible integer value of $c$ for all $a, b > 1$ is

Similar Questions

The number of solutions to the equation $\log _2(x + 5) = 6 - x$ is

If $\frac{1}{\log_3 \pi} + \frac{1}{\log_4 \pi} > x$,then $x$ can be:

$x = \log \left( \frac{1}{y} + \sqrt{1 + \frac{1}{y^2}} \right) \Rightarrow y$ is equal to

If $x = \log_{3} 5$ and $y = \log_{17} 25$,which one of the following is correct?

The solution set of the equation $x^{\log_x(1-x)^2} = 9$ is:

Vedclass Test Series

Exam Paper Generator

Online Exam Module