Q. 25 - Calculus

Question

Now that we have defined $\ln x$ with an integral, we can define a general logarithm with base b as ${\log }_{b}x=\frac{1}{\ln b}\ln x$. In Exercises 24–26 you will investigate this definition of ${\log }_{b}x$.

Use the definition of ${\log }_{b}x$ just given and the new definition of $\ln x$ from Definition 4.36 to express ${\log }_{b}x$ as an integral.

Step-by-Step Solution

Answer

The integral form of the function ${\log }_{b}x$ is ${\log }_{b}x={\int }_1^x\frac{1}{\ln b.\left(t\right)}dt$.

Step 1. Given Information.

The definition:

${\log }_{b}x=\frac{1}{\ln b}\ln x$

${\log }_{b}x$

Step 2. Express the function as integral form.

We know that,

$\ln x={\int }_1^x\frac{1}{t}.dt$

From the definition, ${\log }_{b}x=\frac{1}{\ln b}\ln x$

Substitute $\ln x$ in the above equation,

$$\begin{gathered} {\log }_{b}x=\frac{1}{\ln b}{\int }_1^x\frac{1}{t}.dt \\ {\log }_{b}x={\int }_1^x\frac{1}{\ln b.\left(t\right)}dt \end{gathered}$$