Problem 39 - Algebra for College Students

Question

Use your calculator to find \(x\) when given \(\ln x\). Express answers to five significant digits. $$ \ln x=-3.3244 $$

Step-by-Step Solution

Answer

The value of x is approximately 0.036019.

Step 1: Understanding the Problem

The problem asks us to find the value of \(x\) given that \(\ln x = -3.3244\). The natural logarithm function, \(\ln\), is the inverse of the exponential function with base \(e\). This means that if \(y = \ln x\), then \(x = e^y\).

Step 2: Using the Inverse Function

Since \(\ln x = -3.3244\), we need to calculate \(x = e^{-3.3244}\). Use a calculator that has an \(e^x\) function to find this value.

Step 3: Calculator Use

Enter \(-3.3244\) into your calculator and press the \(e^x\) function to compute \(e^{-3.3244}\).

Step 4: Expressing the Answer

After calculating \(e^{-3.3244}\), record the value to five significant digits as required by the problem.

Key Concepts

Inverse Functions

When we talk about inverse functions, we refer to a function that "reverses" another function. Essentially, if a function takes an input and gives an output, its inverse takes that output and returns it to the original input. In terms of natural logarithms, the inverse function is the exponential function with base \(e\).
This can be expressed as:

• If \(y = \ln x\), then \(x = e^y\).
• This means that when you know \(\ln x\), you can find \(x\) by calculating \(e^{\ln x}\).

Inverse functions are essential for solving equations that involve the use of logarithmic and exponential functions. In problems like the one we are addressing, recognizing the relationship between \(\ln\) and the exponential function allows us to solve for \(x\) when given \(\ln x\).

Exponential Functions

Exponential functions are a type of mathematical function where a constant base is raised to a variable exponent. The most commonly used base in mathematics is the natural constant \(e\), which is approximately 2.71828. This is why functions like \(e^x\) are called exponential functions.

When dealing with an exponential function such as \(x = e^{-3.3244}\), you're using the inverse of the natural logarithm to solve for \(x\). The reason \(e\) is special is due to its unique properties, especially in calculus and continuous growth applications.

• \(e^x\) grows faster than any polynomial.
• It is the base of the natural logarithm, \(\ln\).

Because of these properties, exponential functions are incredibly useful in a wide range of disciplines, from mathematics to natural sciences.

Use of Calculators

Calculators are handy tools for processing complex calculations, especially in situations involving transcendental functions like logarithms and exponentials. To use a calculator correctly for solving exponential problems:

• First ensure that your calculator has an \(e^x\) function.
• Enter the value you want to exponentiate, such as \(-3.3244\).
• Press the \(e^x\) button to get the value of \(e^{-3.3244}\).

The calculator performs the computational heavy-lifting, allowing you to quickly and accurately solve for the desired value of \(x\). Always double-check your input into the calculator to ensure it's accurate, particularly the sign of your exponent.

Significant Digits

The concept of significant digits or figures is crucial in scientific and educational contexts, as it indicates the precision of a calculated or measured quantity. A number's significant digits give an idea of the confidence limits within the measurement or calculation.

• For instance, 0.01052 has five significant digits, indicating precision.
• When expressing answers, especially from calculators, round correctly to the required significant digits, which for this exercise is five.

Rounding and expressing numbers in terms of significant digits can help avoid overstating the precision of an answer. It's important to ensure that your final result, after all calculations are completed, adheres to these requirements to provide a clear, reliable solution. Keep in mind that precision is key in scientific calculations, and adhering to significant figures promotes consistency and uniformity in calculations.