Question

Use a calculator to evaluate each expression. Round your answer to three decimal places. $$ \ln \frac{5}{3} $$

Short answer

0.510

Step-by-step solution

Understand the Expression

The expression given is the natural logarithm of \(\frac{5}{3}\). The natural logarithm function is denoted by \(\text{ln}\) and it represents the logarithm to the base \(e\) where \(e\) is approximately equal to 2.718.

Understand the Property of Logarithm

Recall that the natural logarithm of a fraction \(\frac{a}{b}\) can be written as the difference between the natural logarithm of the numerator and the natural logarithm of the denominator: \(\text{ln} \frac{a}{b} = \text{ln} a - \text{ln} b\).

Using a Calculator

Use the calculator to find the value of \(\text{ln} 5\) and \(\text{ln} 3\). Input these values into the calculator: \( \text{ln} 5 ≈ 1.609 \) and \( \text{ln} 3 ≈ 1.099 \).

Compute the Expression

Subtract the value of \(\text{ln} 3\) from \(\text{ln} 5\). \(\text{ln} \frac{5}{3} = \text{ln} 5 - \text{ln} 3 ≈ 1.609 - 1.099\).

Round to Three Decimal Places

After performing the subtraction, the result is approximately \(0.510 \). Ensure that the answer is rounded to three decimal places.

Key concepts

Logarithmic Properties

Logarithms are mathematical functions that allow us to work with exponential equations more easily. They have some important properties that make them extremely useful. For the natural logarithm, denoted as \(\text{ln}\), here are a few key properties:

• The natural logarithm of a product: \(\text{ln}(a \times b) = \text{ln} a + \text{ln} b\)
• The natural logarithm of a quotient: \(\text{ln} \frac{a}{b} = \text{ln} a - \text{ln} b\)
• The natural logarithm of a power: \(\text{ln}(a^n) = n \times \text{ln} a\)

In our specific exercise, we use the property of logarithms that deals with fractions (quotient rule):

\(\text{ln} \frac{5}{3} = \text{ln} 5 - \text{ln} 3\)

This makes it easier to calculate the logarithm of a fraction by breaking it down into simpler parts. Remembering and utilizing these properties effectively can simplify many logarithmic expressions you may encounter.

Fraction Logarithm

Calculating the natural logarithm of a fraction can seem intimidating at first, but by employing the logarithmic properties, it becomes manageable. Consider the expression \(\text{ln} \frac{5}{3} \). Using the quotient property of logarithms: \(\text{ln} \frac{5}{3} = \text{ln} 5 - \text{ln} 3\). Breaking down the natural logarithm of \(\frac{5}{3}\) this way, transforms it into a simple subtraction problem.
First, find the natural logarithms of the numerator and the denominator separately:

• \(\text{ln} 5 \)
• \(\text{ln} 3 \)

Then, subtract them from each other:

• \(\text{ln} 5 - \text{ln} 3\)

The result provides the natural logarithm of the fraction \(\frac{5}{3}\), making complicated logarithmic calculations more convenient.

Using a Calculator

Using a calculator to find the value of natural logarithms is quite straightforward. Most scientific calculators have a dedicated button for \(\text{ln} \), making the process easier. Here's a step-by-step guide to using your calculator to evaluate the natural logarithm of a fraction:

1. **Turn on your calculator** and make sure it is set to the correct mode for logarithmic calculations.
2. **Input the values**: First, press the \(\text{ln} \) button on your calculator followed by entering the numerator value (in this case, **5**). The screen should display approximately: \(\text{ln} 5 = 1.609\)
3. Press the \(\text{ln} \) button again and then input the denominator value **3**. It should show approximatively: \(\text{ln} 3 = 1.099\)
4. **Perform the subtraction**: Now, subtract the \(\text{ln} 3\) value from \(\text{ln} 5\) to get: \(\text{ln}5 - \text{ln} 3 = 1.609 - 1.099\), which results in **0.510**.
5. **Round the result** to three decimal places, which in our case remains **0.510**.
This method ensures you get the logarithmic values accurately and are able to solve the expressions correctly.