[Math] Show that the equation $\cos(x) = \ln(x)$ has at least one solution on real number

Question

I have question

Q
Show that the equation $\cos (x) = \ln (x)$ has at least one solution on real number.

to solve this question by using intermediate value theorem

we let $f(x)=\cos (x)-\ln (x)$

we want to find $a$ and $b$

but what i should try
to get $f(a)f(b)<0$

I means
$f(a)>0$

$f(b)<0$

thanks

Answer 1

Hint: $\cos$ is bounded whereas $\ln$ is increasing with $\lim\limits_{x\to 0^+} \ln(x) =- \infty$ and $\lim\limits_{x \to + \infty} \ln(x)=+ \infty$.