Problem 21 - Algebra 2 and Trigonometry

Question

In \(21-32,\) for each given logarithm, find \(x,\) the antilogarithm. Write the answer to four decimal places. $$ \ln x=0.5787 $$

Step-by-Step Solution

Answer

The antilogarithm of \(0.5787\) is approximately \(1.7837\).

Step 1: Understand the Problem

We are asked to find the value of \(x\) given that \( \ln x = 0.5787 \). This means we need to find the antilogarithm of \(0.5787\) in the natural logarithm (base \(e\)).

Step 2: Use the Definition of Natural Logarithm

The equation \( \ln x = 0.5787 \) can be rewritten using the definition of a natural logarithm: \( x = e^{0.5787} \).

Step 3: Calculate the Antilogarithm

Calculate \( x \) by raising \( e \) (approximately 2.71828) to the power of \(0.5787\). This will give us the value of \(x\).

Step 4: Perform the Calculation Using a Calculator

Use a calculator to compute \( e^{0.5787} \). Enter \(0.5787\), then use the exponential function to find the value. This should yield \( x \approx 1.7837 \) after rounding to four decimal places.

Key Concepts

Natural Logarithm

The natural logarithm is a fundamental concept in mathematics, often appearing in various equations and problems. It is denoted by \( \ln \) and specifically refers to logarithms that use the base \( e \). Unlike common logarithms that are typically base 10, natural logarithms use the irrational constant \( e \), approximately equal to 2.71828. This makes natural logarithms particularly useful in calculus and mathematical analysis.
To understand it better, when you see \( \ln x \), it fundamentally answers the question: "To what power must we raise \( e \) to get \( x \)?". If \( \ln x = 0.5787 \), it indicates exactly what power of \( e \) results in \( x \).
Natural logarithms feature prominently in growth processes, whether they be biological, radioactive, or financial, due to their continuous compound nature. This makes them a key tool in understanding exponential growth and decay.

Exponential Function

The exponential function is intimately tied to logarithms, particularly the natural logarithm. It is often denoted as \( e^x \), where \( e \) is Euler's number and \( x \) is the exponent. This function is central to both mathematics and applied sciences because it describes processes that change at rates proportional to their current value, a common occurrence in natural phenomena.
In our problem, we faced \( e^{0.5787} \). This expression becomes the connector between logarithmic and exponential worlds. Simply put, it says: "What do we get when we start at 1 and grow continuously by the factor \( e \) over the period equal to \( 0.5787 \)?"
Working on such calculations often requires a calculator due to \( e \)'s irrational nature. Calculators are equipped with functions to handle \( e \) raised to any number. You’ll often find exponential functions used in calculating interest, population growth models, and decay processes.

Base e

In mathematics, \( e \) is known as Euler's number, after the Swiss mathematician Leonhard Euler. It's an irrational number, approximately 2.71828, and is used as the base for natural logarithms. Base \( e \) holds properties that make it extremely valuable, especially in calculus.
One of the key aspects of \( e \) is its role in continuous growth prediction. Whether you're dealing with compound interest, population models, or radioactive decay, \( e \) offers a remarkable insight into natural processes.

• \( e \) defines the limit of \( (1 + \frac{1}{n})^n \) as \( n \) approaches infinity.
• Using \( e \) as a base makes some calculus operations notably simpler.
• The concept of the natural logarithm arises from \( e \), providing a framework for solving equations that involve exponential growth.

Understanding \( e \) opens doors to advanced mathematics and dynamic systems analysis.
Whenever you work with natural growth or decay models, Euler's number becomes an indispensable tool.

Rounding to Decimal Places

Rounding numbers to a specific number of decimal places is a critical skill in mathematics, especially when presenting computed data. It helps in expressing numbers more clearly by limiting the precision to an agreed-upon number of figures, enhancing readability and practicality.
In our exercise, rounding \( x \), calculated as \( e^{0.5787} \), to four decimal places meant presenting our answer as 1.7837. This process involved observing the fifth decimal place to decide whether the fourth decimal should be rounded up or remain.

• If the digit next to the last is 5 or more, increase the last retained decimal by one.
• If it is less, keep the decimal as it is.

Proper rounding is crucial, particularly in scientific calculations and financial applications, where too many decimals can either complicate or mislead the outcomes.
Getting comfortable with rounding practices ensures that mathematical and real-world applications retain their precision without unnecessary complexity.