Problem 44 - Algebra 2

Question

Solve each equation or inequality. \(\ln (x+3)=2\)

Step-by-Step Solution

Answer

\( x \approx 4.389 \)

Step 1: Understand the equation

The given equation is \( \ln(x+3) = 2 \). This is a logarithmic equation where \( \ln \) denotes the natural logarithm, which means it's the logarithm to the base \( e \) (approximately 2.718). The goal is to solve for \( x \).

Step 2: Exponentiate both sides

To eliminate the natural logarithm, raise \( e \) to the power of both sides of the equation. This yields \( e^{\ln(x+3)} = e^2 \). The property \( e^{\ln(a)} = a \) simplifies the left-hand side to \( x+3 \). This gives us the equation \( x+3 = e^2 \).

Step 3: Isolate \( x \)

Subtract 3 from both sides to solve for \( x \). This step gives us \( x = e^2 - 3 \).

Step 4: Calculate \( e^2 \)

Calculate \( e^2 \) using a calculator. Since \( e \approx 2.718 \), \( e^2 \approx 7.389 \).

Step 5: Find the final answer

Substitute the value of \( e^2 \) back into the equation from Step 3 to find \( x \). So, \( x \approx 7.389 - 3 = 4.389 \).

Key Concepts

Natural Logarithms

Natural logarithms use the base \( e \), which is approximately 2.718. This base is special because it arises naturally in various mathematical contexts such as calculus and financial calculations. When you see \( \ln(x) \), it denotes the natural logarithm of \( x \). Unlike common logarithms with base 10, natural logarithms make complex equations more manageable. By converting a multiplication problem into an additive one, it simplifies many calculations.\ To solve an equation like \( \ln(x+3) = 2 \), we utilize the natural logarithm's ability to convert terms into a form where exponentiation can easily reverse the logarithm's effect. This process is a crucial step in understanding how logarithms interact with exponents.

Exponents and Logarithms

Exponentiation and logarithms are inverse operations. To understand this, consider how squaring and square rooting work together. In the same way, an exponent with base \( e \) and the natural logarithm undo each other. This means that \( e^{\ln(a)} = a \) and \( \ln(e^b) = b \). This property is central to solving logarithmic equations.\ In the sample problem, after transforming \( \ln(x+3) = 2 \) into \( e^{2} \) by raising both sides to the power of \( e \), the equation becomes simpler. The base \( e \) and the natural log "cancel" each other out, leading to the straightforward equation \( x+3 = e^2 \). This step shows the beauty of how logarithms and exponents work harmoniously to simplify equations.

Properties of Logarithms

The properties of logarithms are powerful tools that help solve complex mathematical problems. Important features include the product rule, the quotient rule, and the power rule, among others. Each of these helps in breaking down different components of logarithmic expressions.\ For example, the property \( e^{\ln(a)} = a \) is pivotal in solving the exercise in focus. It allows you to transform an equation with a logarithm into an algebraic expression that is much easier to solve. Another key aspect is the understanding that logarithms convert multiplicative relationships to additive ones, making them easier to handle.\ Whether you're addressing the natural logarithm or others, these properties remain fundamental. They assure that any exponential expressions or relationships within the logarithm can be simplified, offering a clear path to solving the equation efficiently and accurately.