Q. 96 - Calculus

Question

Use algebra, limit rules, and the continuity of $\ln x$ on $(0,\infty )$ to prove that every logarithmic function of the form $f\left(x\right)={\log }_{b}x$ is continuous on $(0,\infty )$.

Step-by-Step Solution

Answer

It is proved that every logarithmic function of the form $f\left(x\right)={\log }_{b}x$ is continuous on $(0,\infty )$.

Step 1. Given Information

We are given a function,

$f\left(x\right)={\log }_{b}x$

Step 2. Proving the statement.

Given that $c\in \mathrm{ℝ}$,

$$\begin{gathered} \underset{x\to c}{\lim }f\left(x\right)=\underset{x\to c}{\lim }A{b}^x \\ =A\underset{x\to c}{\lim }{b}^x \\ =A\underset{x\to c}{\lim }{\left(e^{\ln b}\right)}^x \\ =A{\left(\underset{x\to c}{\lim }{e}^x\right)}^{\ln b} \\ =A{\left(e^c\right)}^{\ln b} \\ =A{\left(e^{\ln b}\right)}^c \\ =A{b}^c \\ =f\left(c\right) \end{gathered}$$

Hence, $f\left(x\right)={\log }_{b}x$ is continuous on $(0,\infty )$.