Problem 12 - Intermediate Algebra

Question

The logarithmic equation \(\ln x=1.5318\) is equivalent to the exponential equation _____ = _____.

Step-by-Step Solution

Answer

The logarithmic equation \( \ln x = 1.5318 \) is equivalent to the exponential equation \( e^{1.5318} = x \).

Step 1: Understand the Logarithmic Equation

The logarithmic equation given is \( \ln x = 1.5318 \). The natural logarithm \( \ln x \) means \( \log_e x \), where \( e \) is the base of the natural logarithm and approximately equal to 2.718.

Step 2: Convert to Exponential Form

To convert the logarithmic equation \( \ln x = a \) to an exponential form, rewrite it as \( e^a = x \). Thus, for \( \ln x = 1.5318 \), the equivalent exponential equation is \( e^{1.5318} = x \).

Step 3: Evaluate the Exponential Expression

Calculate \( e^{1.5318} \) to find the actual value of \( x \). Using a calculator, \( e^{1.5318} \approx 4.624 \). Therefore, the exponential equation is \( x = 4.624 \).

Step 4: Write the Final Exponential Equation

Now that we have evaluated the expression, write down the equivalent exponential equation: \( e^{1.5318} = 4.624 \). This shows the relationship in an exponential form.

Key Concepts

Natural Logarithms

Natural logarithms are logarithms that use the number \( e \) as their base. The number \( e \) is an irrational number approximately equal to 2.718 and it is one of the most important constants in mathematics.
When you see \( \ln x \), it’s shorthand for \( \log_e x \). In other words, it is asking "what power must we raise \( e \) to, in order to obtain \( x \)?" Natural logarithms are frequently used in calculus and sciences because they simplify the process of differentiating and integrating exponential functions.
Using natural logs makes it easier to manage exponential behaviors, such as growth and decay, because they directly relate to the exponential function with base \( e \). This property makes them a powerful tool in simplifying mathematical models of real-world phenomena.

Conversion to Exponential Form

To solve problems involving logarithms, sometimes we need to convert the logarithmic form to its equivalent exponential form. This is done when we are given that \( \ln x = a \). We can convert the logarithmic equation into an exponential equation using the property that states \( \ln x = a \) is equivalent to \( e^a = x \).
Understanding this conversion is key in solving for unknowns in logarithmic equations. Here’s how to do it:

• Recognize the equation structure: \( \ln x = a \).
• Transform it by writing \( e \) raised to the power of \( a \): \( e^a = x \).

This conversion is applicable whenever you encounter natural logarithms and need to isolate a variable or express an unknown in exponential terms.

Evaluating Exponential Expressions

Once you have converted a logarithmic equation to its exponential form, the next step is to evaluate the exponential expression. Evaluating\( e^a \) involves calculating the value of \( x \) when it is written as \( x = e^a \). This often requires using a calculator since \( e \) is an irrational number.
Here’s how to do it effectively:

• Enter the exponent value (in our example, it is \( a = 1.5318 \)) into your calculator.
• Use your calculator’s exponential function to compute \( e^{1.5318} \).
• The result will give you the value of \( x \). In this case, \( e^{1.5318} \approx 4.624 \).

Evaluating the expression gives us a numerical understanding of the value of \( x \) that satisfies the original logarithmic equation. This step is crucial for solving real-world applications that utilize exponential growth or decay, providing tangible results from the natural logarithmic relationships.