Problem 54
Question
Use a calculator to find a decimal approximation for each common or natural logarithm. $$\ln 0.014$$
Step-by-Step Solution
Verified Answer
\( \ln 0.014 \approx -4.267 \).
1Step 1: Understanding the Natural Logarithm
The expression \( \ln 0.014 \) represents the natural logarithm of 0.014, which asks the question: "To what power must \( e \) (approximately 2.718) be raised to equal 0.014?". This is the key understanding before we calculate the value.
2Step 2: Using the Calculator
To find \( \ln 0.014 \), enter 0.014 into a calculator. Look for the "ln" function, which represents the natural log. Press "ln" to apply this function to the number.
3Step 3: Calculating the Result
After pressing the "ln" button, the calculator will display the result of \( \ln 0.014 \). Make sure your calculator is in the right mode (usually standard or scientific mode) to perform this calculation accurately.
4Step 4: Interpreting the Result
When you press the “equals” button on the calculator after applying the "ln" function to 0.014, it should display approximately \(-4.267\). This means \( e^{-4.267} \approx 0.014 \).
Key Concepts
Exponential FunctionDecimal ApproximationCalculator Use
Exponential Function
An exponential function is a mathematical expression where a constant base is raised to a variable exponent. The most notable base for exponential functions is Euler's number, denoted as \( e \), which is approximately 2.718. This number is fundamental in calculus and complex analysis due to its unique properties.
When we deal with natural logarithms, we are often solving these exponential equations in reverse. Essentially, we look for the exponent that turns \( e \) into a given number.
For instance, the expression \( e^x \) represents an exponential function, where \( x \) is the exponent. When we're given a value like 0.014 and we need to find what power \( e \) needs to be raised to make this number, we're effectively solving for \( x \) in \( e^x = 0.014 \).
When we deal with natural logarithms, we are often solving these exponential equations in reverse. Essentially, we look for the exponent that turns \( e \) into a given number.
For instance, the expression \( e^x \) represents an exponential function, where \( x \) is the exponent. When we're given a value like 0.014 and we need to find what power \( e \) needs to be raised to make this number, we're effectively solving for \( x \) in \( e^x = 0.014 \).
- Understanding \( e \) is key, as it's the base for natural logs (denoted as \( \ln \)).
- The exponential function \( e^x \) grows rapidly and is used in many natural phenomena.
- The inverse of an exponential function is the logarithm function, specifically the natural log when the base is \( e \).
Decimal Approximation
Decimal approximation is the process of expressing an exact number as a decimal, which may be either finite or infinite, but more usable for practical calculations. In many cases, the exact result of mathematical expressions, like logarithms, may not be easy to denote using simple numbers, hence approximations are used.
The natural logarithm \( \ln 0.014 \) translates to finding an approximation for a very specific number. The exact logarithmic result may be irrational, meaning it cannot be precisely expressed without an infinite series of digits.
For practical purposes:
The natural logarithm \( \ln 0.014 \) translates to finding an approximation for a very specific number. The exact logarithmic result may be irrational, meaning it cannot be precisely expressed without an infinite series of digits.
For practical purposes:
- A calculator helps us approximate \( \ln 0.014 \) to decimal form, giving us approximately -4.267.
- This approximation allows us to easily interpret and use the value without needing to deal with a cumbersome exact expression.
- Decimal approximations are crucial in engineering, finance, and sciences where precise calculations are not possible or necessary.
Calculator Use
Using a calculator effectively can transform complex mathematical problems into simpler tasks. It is an indispensable tool for students, engineers, scientists, and anyone dealing with numbers regularly.
When using a calculator to find a logarithm like \( \ln 0.014 \), follow these steps:
When using a calculator to find a logarithm like \( \ln 0.014 \), follow these steps:
- Enter numerical value: First, type in the number provided, in this case, 0.014.
- Apply the 'ln' function: Locate and press the 'ln' button typically found on scientific calculators. This button corresponds to the natural log function.
- Interpreting the display: After pressing 'ln', the calculator will display the approximate logarithmic value, crucial for understanding the underlying exponential relationship. For \( \ln 0.014 \), a calculator gives roughly -4.267.
Other exercises in this chapter
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