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Chapter 12: Problem 89

Find a formula for converting natural logarithms to common logarithms.

Short Answer

Expert verified
To convert \ln(x) to \log(x), use \log_{10}(x) = \(\frac{\ln(x)}{\ln(10)}\).

Step by step solution

01

Understand the Relationship Between Logarithms

Determine the relationship between the natural logarithm (base e, denoted as \ln) and the common logarithm (base 10, denoted as \log). Recall that for any positive number a, \(a = e^{\ln(a)}\).
02

Apply the Change of Base Formula

Use the change of base formula for logarithms: \log_b(a) = \(\frac{\log_c(a)}{\log_c(b)}\). In this case, to convert \ln(x) to \log(x), set \(a = x\), \(b = 10\) and \(c = e\). This gives us \log_{10}(x) = \(\frac{\ln(x)}{\ln(10)}\).
03

Simplify the Formula

Understand that \ln(10) is a constant. Therefore, the conversion formula from natural logarithms to common logarithms is: \log_{10}(x) = \(\frac{\ln(x)}{\ln(10)}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Natural Logarithm
Natural logarithms are logarithms with the base e. The number e is an irrational number approximately equal to 2.71828. When we write \[ \text{ln}(x) \], it means the exponent to which e must be raised to get x.
For example, \[ \text{ln}(e) = 1 \]. This is because e raised to the power of 1 is e. Natural logarithms are widely used in calculus and mathematical modeling because of their natural growth properties.
Common Logarithm
Common logarithms are logarithms with base 10. These are often used in scientific calculations and data analysis. When we write \[ \text{log}(x) \], it means the exponent to which 10 must be raised to get x.
For example, \[ \text{log}(1000) = 3 \]. This is because 10 raised to the power of 3 is 1000. Common logarithms are helpful for simplifying large numbers and are used in various fields, including engineering.
Change of Base Formula
The change of base formula allows you to convert logarithms from one base to another. The formula is given by \[ \text{log}_b (a) = \frac{\text{log}_c (a)}{\text{log}_c (b)} \]
This formula is useful when you need to switch bases for easier calculations or when using calculators that only handle specific bases. For instance, to convert from natural logarithms to common logarithms, set the base b to 10 and c to e:
\[ \text{log}_{10} (x) = \frac{\text{ln}(x)}{\text{ln}(10)} \]
Base e
Base e is the base of the natural logarithm. The number e is approximately equal to 2.71828 and is an important mathematical constant. It appears in various areas of mathematics, particularly in calculus and complex analysis.
One reason e is special is because of its unique properties in continuous growth models and compound interest calculations. The natural logarithm inherently simplifies problems involving rates of change and exponential growth.
Base 10
Base 10 is the base of the common logarithm. Our number system is also base 10, which makes common logarithms intuitive to use.
Because common logarithms deal with powers of 10, they are practical for simplifying computations with very large or very small numbers. They are also heavily used in scientific notation, where numbers are expressed as multiples of powers of 10 to make them easier to read and understand.

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Most popular questions from this chapter

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