Tasty Logarithm #4

Geometry Level 3

$\large \log_{\sin(x)} \cos (x) +\log_{\cos(x)} \sin(x) =2$

If the minimum value of $x$ that satisfy the equation above is equals to $\dfrac{\pi}{a}$ for some constant $a$ , find the value of $a$ .

Give your answer to 3 decimal places.

Note: Angles are measured in radians.

2 solutions

Parth Lohomi
May 13, 2015

Let $\log_{\sin(x)} \cos (x) =m$

$\therefore$ $\log_{\cos(x)} \sin(x)=\frac{1}{m}$

$\implies$ $m+\frac{1}{m}=2$

$m^2+1=2m\implies (m-1)^2=0,\therefore m=1$

So, $\sin(x) =\cos(x) \implies x\mid_{min} =\frac {\pi} {4}$

So $a=4$ .

Well done using the properties of logarithm. Bonus question: What would be the solution of all $x$ such that the equation is fulfilled?

@Calvin Lin /@mathchallengemaster it would be $n\pi+\frac{\pi} {4}$

Parth Lohomi - 6 years, 1 month ago

This will be incorrect , remember that the for $\log_{a}{b} \quad b\in(0,\infty)-\{1\} \text{ and } a\in (0,\infty)$

Both $\sin{x}$ and $\cos{x}$ should be positive , which means x should lie in the 1st Quadrant

Sabhrant Sachan - 4 years, 9 months ago

Haha We posted identical solutions at the exact same time. No point in leaving mine up any more. :)

Brian Charlesworth - 6 years, 1 month ago

Haha, no problem!!

Parth Lohomi - 6 years, 1 month ago

Noel Lo
May 18, 2015

Interesting problem! Love it!