Problem 38 - Algebra for College Students

Question

Use your calculator to find \(x\) when given \(\ln x\). Express answers to five significant digits. $$ \ln x=-1.6246 $$

Step-by-Step Solution

Answer

x ≈ 0.19734

Step 1: Understand the Problem

We are given the natural logarithm of a number, \( \ln x = -1.6246 \), and we need to find the number \( x \).

Step 2: Use the Antilogarithm

To find \( x \), we need to take the antilogarithm of \( -1.6246 \). This means calculating the exponential \( e \) raised to the given logarithm value: \( x = e^{-1.6246} \).

Step 3: Calculate Using a Calculator

Use a calculator to compute \( x \) by entering \( e^{-1.6246} \). This will give you the value of \( x \) to five significant digits.

Step 4: Express the Solution

Enter the computed value from the calculator to get: \( x \approx 0.19734 \). Ensure it is rounded to five significant figures.

Key Concepts

Antilogarithm

When you're working with logarithms, the term **antilogarithm** often comes up. It's essentially the opposite of taking a logarithm. Imagine you know the logarithm of a number and need to find the original number. That's where antilogarithms come in handy.

The process of finding an antilogarithm involves reversing the logarithmic operation. If you have the natural logarithm (like in our example with \( \ln x = -1.6246 \)), finding the antilogarithm means computing \( e \) raised to that logarithmic value. It can be visualized as:

• Original logarithm given: \( \ln x = y \)
• To find \( x \), compute the antilogarithm: \( x = e^y \)

Antilogarithms are used across many mathematical fields, such as solving exponential growth problems or undoing log transformations in data analysis. Using a calculator to perform these computations is often necessary due to complexity.

Exponential Functions

Exponential functions are a fundamental concept in mathematics, especially when dealing with growth and decay rates.

In essence, an exponential function can be described by the formula \( f(x) = a \cdot e^{bx} \), where \( e \) is Euler's number (approximately 2.71828), \( a \) is a constant that scales the function, and \( b \) dictates the rate of growth or decay.

In our problem, the task was to find the value of \( x \) given its natural logarithm. We used an exponential function in this context by calculating \( e^{-1.6246} \). Here:

• The "\( e \)" part indicates the base of natural logarithms, a constant representing continuous growth.
• The exponent \(-1.6246\) alters the base to produce the required \( x \) value.

Exponential functions appear naturally in various real-world contexts, from population dynamics to financial calculations, making them a crucial mathematical tool.

Significant Figures

Calculating values with accuracy often requires expressing numbers using **significant figures**.

Significant figures are the digits in a number that contribute to its precision. They are crucial when rounding numbers while maintaining accuracy, especially when dealing with precise calculations in science and engineering.

• The first significant figure in a number is the first non-zero digit.
• For example, in 0.19734 (our solution), all digits are considered significant: 1, 9, 7, 3, 4.
• Significant figures help ensure that computed values maintain the desired level of precision.

When a problem specifies a certain number of significant figures (like five in our exercise), it's important to round the final answer accordingly to maintain that precision. This ensures consistency and reliability in mathematical analysis and results.