Problem 61 - Intermediate Algebra

Question

Solve each equation. Express all answers to four decimal places. $$ \ln x=1.4023 $$

Step-by-Step Solution

Answer

\( x = 4.0657 \)

Step 1: Understand the Equation

The equation given is \( \ln x = 1.4023 \). Here, \( \ln \) refers to the natural logarithm, which is the logarithm to the base \( e \), where \( e \approx 2.71828 \). We need to solve for \( x \).

Step 2: Convert to Exponential Form

In the logarithmic equation \( \ln x = 1.4023 \), we can convert this to its exponential form using the property that if \( \ln a = b \), then \( a = e^b \). Therefore, \( x = e^{1.4023} \).

Step 3: Calculate the Exponential Value

To find \( x \), we need to calculate \( e^{1.4023} \). Using a scientific calculator or computational tool, we find this value.

Step 4: Compute Using Calculator

By computing \( e^{1.4023} \) using a calculator, we get approximately 4.0657 when rounded to four decimal places.

Key Concepts

Exponential Equations

Exponential equations are equations where the unknown variable appears as the exponent of a number. A standard form involves an expression such as \( a^x = b \), where \( a \) is the base, \( x \) is the exponent, and \( b \) is the resultant value. These equations are solved by finding the value of the exponent that makes the equation true. Understanding exponential equations is crucial because they appear in various real-world scenarios, such as compound interest calculations, population growth, and radioactive decay.

To tackle exponential equations, remember these points:

• Identify the base of the exponent.
• Isolate the exponential part of the equation, if there are other terms present.
• Use logarithmic operations to solve for the exponent, if necessary.

Converting exponential equations to a logarithmic form can help simplify the problem, making it easier to solve for the unknown variable.

Logarithmic Equations

Logarithmic equations are those that involve logarithms, which are the inverse operations of exponentiation. A simple logarithmic equation has the form \( \log_a b = c \), where \( a \) is the base, \( b \) is the argument, and \( c \) is the logarithm. Solving these equations often requires an understanding of various logarithmic properties, such as converting them into exponential form.

In our example, we have a natural logarithm, \( \ln x = 1.4023 \). This can be rewritten using the property \( \ln a = b \) which implies \( a = e^b \). Therefore, \( x = e^{1.4023} \), turning the logarithmic equation into an exponential equation.

• Always analyze the base of the logarithm to understand its relationship with the argument.
• Convert logarithmic equations to their exponential form to simplify.
• Apply known values and computational tools to find the solution.

Practicing these steps ensures a better grasp on solving logarithmic equations efficiently.

Number e

The number \( e \) is an irrational and transcendental number, meaning it cannot be expressed as a simple fraction and is not a root of any non-zero polynomial equation with rational coefficients. It is approximately equal to 2.71828 and serves as the base of natural logarithms. The number \( e \) is fundamental in mathematics, especially in processes involving growth and decay, like in the fields of calculus, probability theory, and financial modeling.

Key properties of \( e \) include:

• \( e \) arises naturally when calculating compound interest, where it describes continuous growth as interest is added at every moment.
• It is used in calculus, particularly in deriving formulas for exponential growth and decay processes.
• The function \( e^x \) is unique as its derivative is also \( e^x \).

Understanding \( e \) and its applications helps in comprehending exponential and logarithmic functions deeply, aiding in solving complex equations efficiently.