Question

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

Short answer

0.536

Step-by-step solution

Understand the Function

The expression to evaluate is \(\frac{\ln 5}{3}\). This means you need to find the natural logarithm of 5 and then divide the result by 3.

Use a Calculator to Find the Natural Logarithm

On your calculator, locate the natural logarithm function, usually labeled as 'ln'. Enter 5 and press the 'ln' button to get \(\ln 5\). The value should be approximately 1.609.

Divide the Result by 3

Take the value of \(\ln 5 \), which is approximately 1.609, and divide it by 3. Use the division function on your calculator to get \(\frac{1.609}{3} \).

Round to Three Decimal Places

After performing the division, the result is approximately 0.536333. Round this value to three decimal places to get 0.536.

Key concepts

Natural Logarithm

The natural logarithm, denoted as \(\text{ln}\), is a special type of logarithm. It uses the mathematical constant e (approximately 2.718) as its base. This is different from common logarithms that use 10 as their base.
When you see \(\text{ln}(x)\), it is asking you for the power to which e must be raised to result in x. For example, \(\text{ln}(1)\ = 0\), because e raised to the power of 0 is 1. Similarly, \(\text{ln}(e)\ = 1\), because e raised to the power of 1 is e itself.
In the original exercise, we needed to find \(\text{ln}(5)\), which is the power to which e must be raised to get approximately 5. Using a calculator, this value was found to be approximately 1.609.

Logarithmic Functions

Logarithmic functions are the inverses of exponential functions. This means that if you have an exponentiation like \[a^b = c\text{, you can write the equivalent logarithmic form as }\ \text{log}_a(c) = b.\] The base here, a, can be any number, but e is a common and important base in naturallogarithms.
Key properties of logarithmic functions include:

• \(\text{log}_a(1) = 0\)
• \(\text{log}_a(a) = 1\)
• \(\text{log}_a(x \times y) = \text{log}_a(x) + \text{log}_\text{a}(y)\)
• \(\text{log}_a(\frac{x}{y}) = \text{log}_a(x) - \text{log}_a(y)\)

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These properties help in simplifying and solving logarithmic equations and expressions. In our exercise, we used the natural logarithm, which is a specific case of these general logarithmic functions.

Numerical Methods

Numerical methods involve techniques to approximate mathematical procedures. They are especially helpful when a formula doesn’t directly solve the problem or when exact solutions are not possible.
Calculators use numerical methods to compute values like natural logarithms. Specific numerical methods include:

• Newton's Method: Useful for finding successively better approximations to the roots (or zeroes) of a real-valued function.
• Interpolation Methods: Used to estimate unknown values that fall between known values.
• Series Expansion: Natural logarithms can be calculated using series expansions like the Taylor or Maclaurin series.

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In our exercise, the calculator used internal numerical methods to quickly and efficiently find \(\text{ln}(5)\), which we divided by 3 to get the final result.