Problem 35 - College Algebra

Question

For the following exercises, solve for \(x\) by converting the logarithmic equation to exponential form. $$\ln (x)=2$$

Step-by-Step Solution

Answer

The solution is \( x \approx 7.389 \).

Step 1: Identify the Equation Type

The given equation is a natural logarithm equation: \( \ln(x) = 2 \). Here, \( \ln \) denotes the natural logarithm, which is the logarithm to the base \( e \).

Step 2: Convert to Exponential Form

Recall that the natural logarithm \( \ln(x) \) is equivalent to \( \log_e(x) \). By definition of logarithms, if \( y = \ln(x) \), then \( e^y = x \). Thus, the equation \( \ln(x) = 2 \) converts to exponential form as \( e^2 = x \).

Step 3: Calculate the Value

Using a calculator or the approximate value of \( e \) (where \( e \approx 2.71828 \)), calculate \( e^2 \). \( e^2 \approx 7.389 \). Therefore, \( x \approx 7.389 \).

Step 4: Verify the Solution

Substitute \( x = 7.389 \) back into the original logarithmic equation: \( \ln(7.389) \approx 2 \). This confirms our solution is correct, as the left side equates to the right side.

Key Concepts

Exponential Form

Let's dive into how logarithmic equations can be converted to exponential form. Understanding this concept is vital when solving equations involving logarithms, such as the one covered in the original exercise.**What is Exponential Form?** Exponential form is a way of expressing numbers using a base raised to a power. For any logarithmic equation of the form \[ \log_b(x) = y \] it can be rewritten as \[ b^y = x \] in its exponential form.**Converting Logarithms to Exponential Form** The process of converting a logarithmic equation to exponential form is straightforward: - Identify the base of the logarithm.- Express the number as the base raised to the logarithmic equation's result.- This gives insight into the relationship between the number and the base. **Relevance in Solving Equations** In our example with \( \ln(x) = 2 \), the base here is \( e \), the natural constant. Therefore, the exponential equivalent becomes \( e^2 = x \). This transformation simplifies solving for \( x \), as it allows you to directly compute the value using powers.

Natural Logarithm

Natural logarithms are a specific type of logarithm that uses the mathematical constant \( e \) as its base. This makes them appear in various scientific and financial models due to their unique properties.**Understanding Natural Logarithms**
A natural logarithm, denoted by \( \ln(x) \), specifically uses \( e \approx 2.71828 \) as its base. This close association with \( e \) ties them deeply into mathematical calculations and natural growth processes.**Key Properties**- **Inverse Relationship**: The natural logarithm is the inverse of the exponential function with base \( e \). - **Simplification**: For \( y = \ln(x) \), it directly implies \( e^y = x \), allowing easy transitions between these forms. - **Log Identity**: Another useful identity is that \( \ln(1) = 0 \) and \( \ln(e) = 1 \), forming the base for more complex mathematics.**Why Use Natural Logarithms?**
Natural logarithms simplify the process of working with exponential growth problems, making them ideal for fields like biology and finance. By converting equations using \( \ln \), such as \( \ln(x) = 2 \) into \( e^2 = x \), computational tasks become more manageable.

Base e

When we encounter logarithmic and exponential equations, the constant \( e \), often known as Euler's number, frequently appears as the base. The importance of \( e \) arises from its natural occurrence in continuous growth scenarios.**Unpacking Base \( e \)**
The number \( e \) is an irrational constant approximately equal to 2.71828. It is the foundation of natural exponential functions, showing up in growth patterns across nature and finance. **Properties of Base \( e \)**- **Irrational Nature**: Being irrational, \( e \) cannot be exactly represented as a fraction. - **Exponential Growth**: Functions of the type \( e^x \) represent continuous exponential growth, making \( e \) critically valuable in modeling real-world changes.- **Rate of Change**: Unique to \( e \), its properties include a rate of increase proportional to its current value.Functions involving \( e \) allow us to model and solve complex equations more efficiently. For instance, transforming \( \ln(x) = 2 \) into \( e^2 = x \) lets us utilize the base's properties to quickly calculate \( x \) using known values or computational tools.