Problem 27 - Algebra 2 and Trigonometry

Question

In \(21-32,\) for each given logarithm, find \(x,\) the antilogarithm. Write the answer to four decimal places. $$ \ln x=4.8200 $$

Step-by-Step Solution

Answer

The antilogarithm is approximately 124.8709.

Step 1: Recall the Relationship Between Logarithms and Exponentials

The equation given is \( \ln x = 4.8200 \). The natural logarithm \( \ln x \) is the logarithm to the base \( e \), where \( e \) is approximately 2.71828. Thus, \( \ln x = 4.8200 \) can be rewritten in exponential form: \( x = e^{4.8200} \).

Step 2: Calculate the Antilogarithm Using Exponentiation

To find \( x \), we need to compute \( e^{4.8200} \). This requires using the exponential function with base \( e \). Using a calculator, raise \( e \) to the power 4.8200. This gives \( x \approx 124.8709 \).

Step 3: Round to Four Decimal Places

Ensure that the calculated value of \( x \) is rounded to four decimal places. The value \( 124.8709 \) becomes \( 124.8709 \) when rounded to four decimal places.

Key Concepts

Antilogarithm

The antilogarithm is a way to express the inverse operation of taking a logarithm. In other words, if you have a logarithmic equation like \( \log_b(x) = y \), the antilogarithm will help you find the original number \( x \) by reversing the logarithm.

• Logarithms break complex equations into manageable multiplications.
• However, finding antilogarithms brings us back to the original value or helps solve the exponential expression.

For example, if you know \( \ln x = 4.8200 \), then the antilogarithm process will tell us that \( x = e^{4.8200} \). Here, \( e \) is the base of the natural logarithm. Using a calculator to compute \( e^{4.8200} \), we discover that the antilogarithm is approximately 124.8709, when rounded to four decimal places. This process is crucial in mathematical applications where the exponential growth or decay needs to be reversed into more familiar numbers.

Natural Logarithm

A natural logarithm is a logarithm to the base \( e \), where \( e \approx 2.71828 \). It is represented by "\( \ln \)" and is widely used in calculus and exponential growth problems because it simplifies the differentiation and integration process.

• Natural logarithms are expressed as \( \ln(x) \), where \( x \) is the number you wish to find the logarithm of.
• The natural logarithm of 1 is 0, since \( e^0 = 1 \).

Natural logarithms provide insights into how quantities change over time, making them especially important in sciences such as biology, chemistry, and economics. In the exercise example, \( \ln x = 4.8200 \) signifies a relationship between \( x \) and \( e \), which is pivotal for uncovering exponential expressions by removing the logarithmic function.

Exponential Function

An exponential function takes the form \( f(x) = a \cdot b^x \), where \( a \) is a constant, \( b \) is a positive number, and \( x \) is the exponent. In the context of natural logarithms and antilogarithms, we focus on the natural base \( e \), so the typical form is \( f(x) = e^x \).

• The base \( e \) arises naturally in processes involving growth and decay, such as population growth or radioactive decay.
• Exponentials have unique properties, such as growth rates proportional to their current value.

In our step-by-step solution, \( x = e^{4.8200} \) is used to find the value corresponding to the natural logarithm \( \ln x \). Calculating \( e^{4.8200} \) illustrates how quickly functions can grow. The exponential function underpins much of modern applied mathematics, providing crucial insight into complex dynamic systems. By knowing these functions, you can effectively model and predict various real-world phenomena.