Question

prove that the function g(x)=ln(x)-1/4 has a solution in the interval (0,5)

Answer 1

The function $g(x)=\ln (x)-\dfrac14$ is considered. The function $g$ is continuous in $[1,5]$ because it is a sum of continuous functions. Moreover, it changes sign at the extremes of the interval:

$$g(1)=\ln (1)-\dfrac14=0-\dfrac14=-\dfrac14<0.$$

$$g(5)=\ln (5)-\dfrac14>0.$$

Therefore, by Bolzano's theorem, there exists $c\in (1,5)$ , such that $g(c)=0$ , which proves that $g$ has a zero in this interval, and therefore, also in the interval $(0,5)$ because $g$ is well-defined in this interval.