Problem 66 - Elementary and Intermediate Algebra

Question

Solve each equation. Express all answers to four decimal places. See Example 5. $$ \ln x=-0.28 $$

Step-by-Step Solution

Answer

The solution is approximately 0.7569.

Step 1: Understand the Equation

We are given the equation \( \ln x = -0.28 \). This equation uses the natural logarithm, \( \ln \), which is the logarithm to the base \( e \), where \( e \) is approximately equal to 2.71828.

Step 2: Eliminate the Natural Logarithm

To solve for \( x \), we need to eliminate the natural logarithm by exponentiating both sides of the equation. This means we must raise \( e \) to the power of both sides.The equation becomes: \( x = e^{-0.28} \)

Step 3: Calculate Exponentiation

Now, let's calculate the value of \( e^{-0.28} \). Using a calculator, we find:\[ e^{-0.28} \approx 0.7569 \]

Step 4: Present the Solution

The solution to the equation \( \ln x = -0.28 \) is \( x \approx 0.7569 \). We round to four decimal places as required.

Key Concepts

Understanding Natural Logarithms

Natural logarithms are fundamental in mathematics and are denoted by \( \ln \). They are logarithms that use the base \( e \), which is an irrational constant approximately equal to 2.71828. Natural logarithms are widely used in calculus and natural sciences because they simplify the integration and differentiation of exponential functions.
In practical terms, a natural logarithm can be understood as the power to which \( e \) must be raised to get a particular number. For example, if you have \( \ln x = y \), it means \( e^y = x \). Understanding this relationship helps us to convert between logarithmic and exponential forms easily.
Working with natural logarithms requires familiarity with their properties, such as:

• \( \ln 1 = 0 \)
• \( \ln e = 1 \)
• \( \ln(mn) = \ln m + \ln n \)
• \( \ln(m/n) = \ln m - \ln n \)
• \( \ln(m^n) = n \ln m \)

These properties allow for efficient manipulation of logarithmic expressions, making them more approachable.

The Power of Exponentiation

Exponentiation is the process of raising a number to a certain power. In the context of solving exponential equations, exponentiation is used to "undo" the effect of a logarithm and retrieve the original number. For instance, in the equation \( \ln x = -0.28 \), we use exponentiation to express it as \( x = e^{-0.28} \). Here, raising \( e \) to the power of \(-0.28\) allows us to solve for \( x \).
Using a calculator becomes handy when evaluating complex exponentials like \( e^{-0.28} \). The result approximately equals 0.7569, illustrating how exponentiation converts the logarithmic form back into a more straightforward numerical answer.
Exponentiation also has distinct properties:

• \( a^0 = 1 \) for any \( a \) except 0
• \( a^1 = a \)
• \( a^{m+n} = a^m \cdot a^n \)
• \( a^{m-n} = \frac{a^m}{a^n} \)
• \( (a^m)^n = a^{m\cdot n} \)

Understanding these properties makes working with exponents much easier and intuitive.

Rounding Decimals with Precision

Rounding decimals is an essential skill in mathematics, especially when approximate values are required, as is the case with many equations and scientific calculations. Rounding to a certain number of decimal places means adjusting a number to make it easier to read or interpret, while maintaining a required level of precision.
In our solution, rounding is used to present \( x \approx 0.7569 \) to four decimal places. Here's how you can round to a specific number of decimal places:

• Identify the decimal place to which you want to round. For four decimal places, look at the fifth decimal place.
• If the digit in the next decimal place (fifth here) is 5 or more, increase the fourth decimal place by 1.
• If the digit is less than 5, leave the fourth decimal place unchanged.

In this case, since the value calculated is approximately 0.756898, you would round it to 0.7569 by checking the fifth decimal place (8), which is more than 5.
Being meticulous with rounding helps ensure clarity and correctness in mathematical results, particularly in scientific and financial computations.