Problem 24 - Algebra 2 and Trigonometry

Question

In \(21-32,\) for each given logarithm, find \(x,\) the antilogarithm. Write the answer to four decimal places. $$ \ln x=1.7790 $$

Step-by-Step Solution

Answer

\( x \approx 5.9259 \)

Step 1: Understanding the Problem

We are given the natural logarithm of a number, \( \ln x = 1.7790 \). Our task is to find \( x \), which is the antilogarithm of this value.

Step 2: Converting from Natural Logarithm

The antilogarithm in this context means finding \( x \) such that \( x = e^{1.7790} \), where \( e \) is the base of natural logarithms (approx. 2.71828).

Step 3: Calculation Using Exponential Function

To solve for \( x \), calculate the exponential function: \( x = e^{1.7790} \). Use a calculator you input \( e^{1.7790} \) to find \( x \).

Step 4: Rounding to Four Decimal Places

After calculating \( e^{1.7790} \), we get \( x \approx 5.9259 \). Ensure to round and present this number to four decimal places.

Key Concepts

Understanding Natural Logarithms

Natural logarithms are logarithms with the base "e," a mathematical constant approximately equal to 2.71828. The notation for natural logarithms is "\( \ln x \)," and it represents the power to which \( e \) must be raised to equal \( x \). For example, if \( \ln x = 1.7790 \), it means \( e^{1.7790} = x \).
Natural logarithms are used in many fields such as mathematics, physics, and engineering, helping to model growth processes, calculate compound interest, and solve equations involving exponential growth or decay. Understanding this concept is key to solving problems involving exponential equations with the natural logarithm.

• It helps to simplify multiplication into addition by changing the form of numbers
• Used widely in differentiating and integrating functions

Grasp the idea that \( \ln x \) is a measurement of the rate of growth involved when something continuously grows according to its proportionality constant \( e \). This grasp can be very helpful in practical contexts, enabling straightforward calculations in real-world applications.

Understanding the Exponential Function

An exponential function involving the natural base \( e \) is expressed as \( e^x \). This function is crucial in reversing natural logarithms, as exponentiating a natural logarithm will return you to the original number you started with.
The function has a unique property where the rate of increase of the function \( e^x \) is equal to its current value, mirroring many natural growth processes.

• The exponential function is continuous and defined for all real numbers
• This function grows extremely fast as the value of \( x \) increases

When you calculate \( e^{1.7790} \), you're finding the point along the curve \( y = e^x \) that corresponds to an \( x \)-value of 1.7790. This involves calculating the value of \( x \) such that the original relationship defined by the logarithm \( \ln(x) = 1.7790 \) is maintained.

Rounding to Four Decimal Places

Rounding a number to four decimal places means adjusting the number to the fourth digit following the decimal point. This practice is integral when presenting precise results in scientific and mathematical calculations.
Consider an exponential result such as \( e^{1.7790} \) which yields approximately \( 5.925895 \). To round to four decimal places, observe the fifth digit after the decimal, which in this case is a "9" following "5.9258".
When the fifth decimal place is 5 or higher, increase the fourth decimal place by one. Thus, \( 5.925895 \) becomes \( 5.9259 \).

• Ensures uniformity and precision in reporting numerical data
• Helps in achieving consistency across various calculations and data representations

This method fosters consistency and accuracy in your results, crucial when comparing or aggregating data across different computations or reporting metrics.