Chapter 1: Q. 96 (page 137)

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

Short Answer

Expert verified

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

Step by step solution

Step 1. Given Information

We are given a function,

$f (x) = \log_{b} x$

Step 2. Proving the statement.

Given that $c \in ℝ$ ,

$\lim_{x \to c} f (x) = \lim_{x \to c} A b^{x} = A \lim_{x \to c} b^{x} = A \lim_{x \to c} {(e^{\ln b})}^{x} = A {(\lim_{x \to c} e^{x})}^{\ln b} = A {(e^{c})}^{\ln b} = A {(e^{\ln b})}^{c} = A b^{c} = f (c)$

Hence, $f (x) = \log_{b} x$ is continuous on $(0, \infty)$ .

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

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Proving the statement.

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Use algebra, limit rules, and the continuity of lnxon (0,∞)to prove that every logarithmic function of the form f(x)=logbx is continuous on(0,∞).

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Proving the statement.

One App. One Place for Learning.

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