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Chapter 3: Problem 32

Given \(\ln 4=1.3863\) and \(\ln 5=1.6094,\) find each value. Do not use a calculator. $$\ln \frac{4}{5}$$

Short Answer

Expert verified
\frac{4}{5} = -0.2231

Step by step solution

01

Recall the logarithm property

To solve \( \frac{4}{5}\), use the properties of logarithms. Specifically, use the quotient rule: \( \frac{a}{b} = a - b \).
02

Substitute the given values

We know \( 4 = 1.3863\) and \( 5 = 1.6094\). Substitute these values into the equation from Step 1.
03

Perform the subtraction

Substitute the known values into the quotient rule expression: \( \frac{4}{5} = 4 - 5\). This becomes \( \frac{4}{5} = 1.3863 - 1.6094\).
04

Calculate the result

Perform the subtraction: \( \frac{4}{5} = 1.3863 - 1.6094 = -0.2231\).

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Key Concepts

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

natural logarithm
The natural logarithm, often denoted as \(\backslash ln\), is a logarithm with the base \(e\), where \(e\) is approximately equal to 2.71828. Natural logarithms are used to solve equations involving exponential growth or decay. They provide a convenient way to work with growth rates, compound interest calculations, and more. For instance, in our given problem \(\ln 4 = 1.3863\) and \(\ln 5 = 1.6094\). This means that \(e^{1.3863} \approx 4\) and \(e^{1.6094} \approx 5\).
logarithm properties
To solve logarithmic equations, it is important to be familiar with various properties of logarithms. Here are a few key properties:
  • Product Rule: \(\ln(ab) = \ln(a) + \ln(b)\)
  • Quotient Rule: \(\ln(\frac{a}{b}) = \ln(a) - \ln(b)\)
  • Power Rule: \(\ln(a^b) = b \cdot \ln(a)\)
These properties make it easier to simplify logarithmic expressions and solve logarithmic equations. In our exercise, the quotient rule is particularly useful.
quotient rule for logarithms
The quotient rule for logarithms states that the logarithm of a quotient is equal to the difference of the logarithms of the numerator and the denominator. Mathematically, it鈥檚 expressed as \(\ln(\frac{a}{b}) = \ln(a) - \ln(b)\). This rule helps to break down complex fractions into simpler terms, making it easier to compute. In our exercise, we used the quotient rule to find \(\ln(\frac{4}{5})\). By applying this rule, we simplified the expression to \(\ln(4) - \ln(5)\).
logarithmic subtraction
Logarithmic subtraction involves subtracting one logarithm from another. Using the quotient rule, we can see that \(\ln(\frac{4}{5})\) becomes \(\ln(4) - \ln(5)\). Given the values \(\ln(4) = 1.3863\) and \(\ln(5) = 1.6094\), we subtract to find: \[\begin{align*} \ln(\frac{4}{5}) &= \ln(4) - \ln(5) \ &= 1.3863 - 1.6094 \ &= -0.2231 \end{align*}\]. This example shows how logarithmic properties simplify the calculation, providing a clear and structured solution.

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Most popular questions from this chapter

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