Chapter 3.4, Problem 69E

Solution

To solve: A logarithm inequality.

Conjecture states: In a right angle triangle the difference of squares of tangent (tan) and secant (sec) of the angle equals 1.

Given data: The given inequality is ${1.2}^x\le {\log }_{1.2}x$ .

Method/Formula used:

If $a^m=p$ (index form) can be written as ${\log }_{a}p=m$ . The change of base of log formula is

  ${\log }_{b}a=\frac{{\log }_{e}a}{{\log }_{e}b}$

Calculations:

The given inequality is,

  ${1.2}^x\le {\log }_{1.2}x$ … (1)

The right side of the inequality can be written as:

  ${\log }_{1.2}x=\frac{\ln x}{\ln 1.2}$

Therefore, the inequality (1) becomes,

  ${1.2}^x\le \frac{\ln x}{\ln 1.2}$

The graphs of equations $y={1.2}^x$ , $y={\log }_{1.2}x$ and the inequality ${1.2}^x\le {\log }_{1.2}x$ are shown in Fig. 1.

  

From the graphs of $y={1.2}^x,y={\log }_{1.2}x$ , and inequality ${1.2}^x\le {\log }_{1.2}x$ , it is clear that the solution of the inequality ${1.2}^x\le {\log }_{1.2}x$ is $1.258\le x\le 14.67$ .