Problem 37 - Algebra and Trigonometry

Question

Solve the logarithmic equation for \(x.\) \(\ln x=10\)

Step-by-Step Solution

Answer

\(x \approx 22026.47\)

Step 1: Understand the Problem

We are given the logarithmic equation \( \ln x = 10 \). Our goal is to solve for \( x \). The symbol \( \ln \) represents the natural logarithm, which is the logarithm to the base \( e \) (where \( e \approx 2.71828 \)).

Step 2: Transform to Exponential Form

To solve the equation \( \ln x = 10 \), we can convert the logarithmic form to its equivalent exponential form. The expression \( \ln x = 10 \) means that the logarithm of \( x \) with base \( e \) is 10. We can rewrite this as an exponential equation: \( x = e^{10} \).

Step 3: Calculate the Value of x

Using a calculator, we evaluate \( e^{10} \). The calculation gives us \( x \approx 22026.4657948 \). Therefore, the value of \( x \) that satisfies the equation \( \ln x = 10 \) is roughly 22026.47.

Key Concepts

Natural Logarithm

The natural logarithm, denoted as \(\ln\), is a special type of logarithm that uses the number \(e\) as its base. The number \(e\) is an irrational constant approximately equal to 2.71828. This constant is important in mathematics, particularly in calculus, as it frequently appears in problems involving rates of growth or decay.
When we see \(\ln x\), it means that we are looking for the power to which \(e\) must be raised to get \(x\). Hence, \(\ln x = y\) can be interpreted as saying that \(e^y = x\). The natural logarithm is the inverse function of the exponential function with base \(e\). This relationship is crucial for reverting exponential equations back to linear ones, which is often necessary for solving them.

Exponential Form

Converting a logarithmic equation to exponential form is a pivotal part of solving natural logarithm equations. When you have an equation like \(\ln x = 10\), this can be translated into the exponential expression \(x = e^{10}\). This form makes it easier to solve because it uses basic exponent rules rather than logarithms.
To transform \(\ln x = y\) to its exponential form involves recognizing that \(\ln x\) is asking the question: *What power of \(e\) gives \(x\)?* Therefore, the equation \(x = e^y\) directly tells us the value of \(x\) by raising \(e\) to the power of \(y\). This conversion is essential when solving equations involving natural logarithms because it provides a straightforward path to find the unknown variable.

Solving Equations

Solving equations involving natural logarithms often comes down to converting them into exponential form. In our case, we changed \(\ln x = 10\) into \(x = e^{10}\). The next step involves calculating \(e^{10}\). Using a calculator is typically necessary for this due to the complexity of the calculation.
Once we solve for \(x\), we ensure that \(x\) is positive since logarithms of negative numbers or zero are undefined. By calculating \(e^{10}\), we find that \(x\) approximately equals 22026.47. It's important to have a calculator handy for these computations, as they rely on precision.

• Start by identifying the form of the equation.
• Convert logarithmic form to exponential for simplicity.
• Calculate the result using a calculator or software.

This method streamlines the process of finding solutions to many logarithmic problems.