Q. 24 - 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 ${\log }_{b}x$.

For which b does the definition of ${\log }_{b}x$ just given make sense, and why? Is this the same allowable range of values for b that we saw in our old definition of ${\log }_{b}x$ from Chapter 0?

Step-by-Step Solution

Answer

The value of b for which the definition ${\log }_{b}x=\frac{1}{\ln b}\ln x$ makes sense is $b=e$.

Step 1. Given Information.

The function:

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

Step 2. Find the value of b.

Put $b=e$ such that,

$\frac{1}{{\log }_{e}b}=\frac{1}{\ln b}; {\log }_{e}x=\ln x$ so that,

The definition makes sense when $b=e$.