Problem 57 Explain why $$ \ln x \approx... [FREE SOLUTION]

91影视

Precalculus A Prelude to Calculus

Sheldon Axler

$Math Studyset 91影视 Explanations$ Math

2 Edition

Chapter 3: Problem 57

Explain why $$ \ln x \approx 2.302585 \log x $$ for every positive number $x$.

Short Answer

Expert verified

Using the change of base formula, we can relate natural logarithm ($\ln x$) and common logarithm ($\log x$) as $\ln x = \frac{\log_{10}(x)}{\log_{10}(e)}$. Calculating the value of $\log_{10}(e) \approx 0.4342945$, we can approximate the relation as $\ln x \approx \frac{1}{0.4342945}\log_{10}(x) \Rightarrow \ln x \approx 2.302585\log_{10}(x)$ for every positive number $x$.

Step by step solution

Understanding natural logarithm and common logarithm

The natural logarithm ($\ln x$) is the logarithm to the base $e$, where $e$ is an irrational number approximately equal to 2.71828. The common logarithm ($\log x$) is the logarithm to the base 10, which is widely used in science and engineering for simplifying calculations involving powers of 10. Our task is to show that $\ln x \approx 2.302585 \log x$ for every positive number $x$.

Making use of the change of base formula

The change of base formula allows us to express a logarithm with an arbitrary base in terms of logarithms with another base. The change of base formula is given as follows: \[\log_{a}(x) = \frac{\log_{b}(x)}{\log_{b}(a)}\] In our case, we want to find an approximate relation between $\ln x$ and $\log x$. So we can use the change of base formula with $a = e$ and $b = 10$: \[\ln x = \log_e(x) = \frac{\log_{10}(x)}{\log_{10}(e)}\]

Calculating the value of $\log_{10}(e)$

To show that $\ln x \approx 2.302585 \log x$, we need to find the value of $\log_{10}(e)$. Using a calculator or referring to a logarithmic table, we can find this value: \[\log_{10}(e) \approx 0.4342945\]

Finding the approximate relation between $\ln x$ and $\log x$

We know that $\ln x = \frac{\log_{10}(x)}{\log_{10}(e)}$, and we have the value of $\log_{10}(e) \approx 0.4342945$. Now we can rearrange this equation to find the approximate relation between $\ln x$ and $\log x$: \[\ln x \approx \frac{1}{0.4342945} \log_{10}(x)\] Finally, calculating the value of the constant: \[\frac{1}{0.4342945} \approx 2.302585\] So, we have: \[\ln x \approx 2.302585 \log_{10}(x)\] This shows the approximate relationship between the natural logarithm ($\ln x$) and the common logarithm ($\log x$) for every positive number $x$.

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91影视

Explain why $$ \ln x \approx 2.302585 \log x $$ for every positive number \(x\).

Short Answer

Step by step solution

Understanding natural logarithm and common logarithm

Making use of the change of base formula

Calculating the value of \(\log_{10}(e)\)

Finding the approximate relation between \(\ln x\) and \(\log x\)

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