Problem 46 - Calculus and its applications

Question

Find each logarithm. Round to six decimal places. $$ \ln 0.00087 $$

Step-by-Step Solution

Answer

\( \ln 0.00087 \approx -7.043189 \)

Step 1: Understand the Logarithm Function

The notation \( \ln x \) refers to the natural logarithm of \( x \), which is the logarithm to the base \( e \). In this problem, we need to find \( \ln 0.00087 \).

Step 2: Estimate the Value

First, we recognize that \( 0.00087 \) is a number less than 1, which means its natural logarithm will be negative. This is because the function \( \ln(x) \) for any \( 0 < x < 1 \) produces negative values.

Step 3: Use a Calculator for Precision

Since manual calculations for logarithms are complex, use a scientific calculator to find \( \ln 0.00087 \). Enter 0.00087 and press the natural logarithm function button (often labelled 'ln') on the calculator.

Step 4: Round the Result

The calculator returns a value which you'll round to six decimal places to get \( \ln 0.00087 \approx -7.043189 \).

Step 5: Answer Verification

Verify the result by checking whether rounding or calculator input mistakes affected your final answer. The calculation should consistently give \( -7.043189 \) when repeated on a reliable calculator.

Key Concepts

Logarithmic Calculations

Logarithmic calculations are a fundamental aspect of mathematics used to determine the exponent needed for a specific base to arrive at a given number. In our exercise, the base is the natural number \( e \), approximately equal to 2.71828. Understanding this concept helps you compute the value of expressions like \( \ln x \), where \( \ln \) denotes a logarithm base \( e \). For any positive number \( x \), \( \ln x \) indicates the power one must raise \( e \) to get \( x \).
\[ \ln x = y \quad \text{means} \quad e^y = x \]
When dealing with numbers less than 1, as in this case with \( 0.00087 \), you should expect a negative result, since the number is smaller than our base. This is a key insight while doing logarithmic calculations as it provides insights into the expected sign (positive or negative) of your answer. Furthermore, it's essential to utilize precise tools or calculators since logarithms can have highly specific values that are crucial in various practical applications, such as exponential decay and growth models.

Scientific Calculator

A scientific calculator is an essential tool for mathematics, especially for complex calculations like those involving logarithms. It's equipped with functions to ease the process of finding the natural log, depicted as 'ln' on most calculators.
Here's a simplified way of using a scientific calculator to find \( \ln 0.00087 \):

• Power on your calculator and ensure it's in standard operation mode.
• Enter "0.00087" on the keypad.
• Locate and press the 'ln' button for the natural logarithm function.
• The screen should display about -7.043189, accurate to six decimal places.

Using a calculator minimizes error and provides precision that's challenging to achieve manually. Remember to double-check your input and ensure that you are rounding the result accurately for tasks requiring high precision.

Natural Log Properties

The natural logarithm, or \( \ln \), has several properties that make it unique and useful, especially in calculus and applied mathematics.

• Inverse Function: The natural logarithm is the inverse of the exponential function \( e^x \). This means if \( \ln(x) = y \), then \( e^y = x \).


• Domain: \( \ln(x) \) is only defined for \( x > 0 \). It doesn't exist for zero or negative numbers because there's no real number \( y \) such that \( e^y \) results in zero or a negative.


• Logarithm of a Product: \( \ln(ab) = \ln(a) + \ln(b) \). This property simplifies multiplication into addition, being extremely helpful in simplifying expressions.


• Logarithm of a Quotient: \( \ln(a/b) = \ln(a) - \ln(b) \), helping to break down divisions into subtraction operations.


• Logarithm of a Power: \( \ln(a^b) = b \cdot \ln(a) \), which converts powers to a more manageable multiplication format.

Understanding these properties enhances your ability to manipulate and solve logarithmic equations efficiently, playing a vital role in advanced mathematics and real-world problem-solving.