Problem 23 - Algebra 2 and Trigonometry

Question

In \(21-32,\) for each given logarithm, find \(x,\) the antilogarithm. Write the answer to four decimal places. $$ \ln x=1.3826 $$

Step-by-Step Solution

Answer

The antilogarithm is approximately 3.9860.

Step 1: Understand the Problem

We're given the natural logarithm of a number, \( \ln x = 1.3826 \), and need to find \(x\). The antilogarithm of a natural log is found using the exponential function.

Step 2: Recall the Exponential Property

Remember that if \( y = \ln x \), then \( x = e^y \). Here, \( y = 1.3826 \).

Step 3: Compute the Antilogarithm

Calculate \( x \) using \( x = e^{1.3826} \). Use a calculator to find that \( e^{1.3826} \approx 3.9860 \).

Step 4: Rounding

Round the answer \( 3.9860 \) to four decimal places, which it already is.

Key Concepts

Antilogarithm

An important concept to understand in mathematics, particularly in the study of logarithms, is that of an antilogarithm. When we take the logarithm of a number, we are essentially determining the power or exponent that yields this number when applied to a specific base. The antilogarithm, therefore, is the process of reversing this operation.
If we consider a natural logarithm, such as the one in our exercise, \(\ln x = 1.3826\), the antilogarithm of this value will be the actual number \(x\). This is done by transforming the logarithmic equation into an exponential form. In simpler terms, it means finding which power of base \(e\), Euler's number (approximately equal to 2.71828), gives us the result of \(x\).

• For any natural logarithm \(\ln x = y\), the antilogarithm is computed as \(x = e^y\).
• In this exercise, to find \(x\), we compute the antilogarithm of 1.3826 by computing \(e^{1.3826}\).

Understanding antilogarithms is crucial in converting logarithmic values back to their original form in various scientific calculations.

Exponential Function

The exponential function is one of the most fundamental functions in mathematics. It is denoted by the expression \(e^x\), where \(e\) is the constant approximately equal to 2.71828. This function rapidly increases as the value of \(x\) increases, representing a consistent rate of growth.
The exponential function's relationship with natural logarithms is pivotal. If \(y = \ln x\), then expressing \(x\) as \(e^y\) allows us to 'undo' the logarithmic operation. This link makes the exponential function essential in solving any problem involving natural logarithms.

• The beauty of the exponential function in this context is its ability to translate logarithmic expressions back to their original values, granting us a numerical value of \(x\) when we know \(\ln x\).
• In our solution, we find \(x\) by calculating \(e^{1.3826}\).

Mastering the exponential function is key because it helps us navigate problems spanning across finance, engineering, and natural sciences where such logarithmic expressions often occur.

Rounding to Decimal Places

When working with logarithmic and exponential calculations, precision is crucial. Rounding is a key skill that ensures our results are both accurate and presentable. Often, answers need to be rounded to a certain number of decimal places, providing a cleaner and more manageable result.
The exercise features a solution rounded to four decimal places, which involves assessing the fifth decimal digit to decide whether to round up or down. Rounding follows a simple rule:

• If the next digit after your desired decimal place is 5 or greater, you round up.
• If it is less than 5, you round down.

For example, in this exercise, we calculated \(e^{1.3826} \approx 3.986024\). The answer is succinctly reported as 3.9860 because the fifth digit, 2, is less than 5.
Understanding how to properly round numbers is important in reporting results in any scientific or mathematical field, ensuring that answers are accurate yet practical.