Problem 57 - Algebra 2

Question

Solve each equation. $$ 4 e^{x+2}=32 $$

Step-by-Step Solution

Answer

The solution to the equation \(4 e^{x+2} = 32\) is \(x = \ln(8) - 2\).

Step 1: Transform to base-form

The equation is \(4 e^{x+2} = 32\). This can be rewritten as \(e^{x+2} = 8\), as 32 divided by 4 equals 8.

Step 2: Apply natural logarithm

Then, apply the natural logarithm to both sides resulting in \(\ln(e^{x+2}) = \ln(8)\). According to logarithm rules, this simplifies to \((x+2)\ln(e) = \ln(8)\). Here, \(\ln(e)\) is equal to 1, so this equation simplifies to \(x+2 = \ln(8)\).

Step 3: Solve for X

Finally, solve for x by subtracting 2 from both sides of the equation. This gives the solution \(x = \ln(8) - 2\).

Key Concepts

Logarithms

Logarithms are a crucial mathematical tool used to solve equations involving exponentials. Essentially, a logarithm is the inverse operation of exponentiation. If you know the base and the exponent, you can determine the result. Conversely, if you know the result and the base, the logarithm helps find the exponent.

Here's what you need to know about logarithms:

• The logarithm of a number is the exponent to which the base, usually 10 or e, is raised to produce that number.
• The equation "\(b^y = x\)" can be rewritten in its logarithmic form as "\(\log_b(x) = y\)".
• Logarithms have specific properties that make them particularly useful in solving equations, such as \(\log_b(mn) = \log_b(m) + \log_b(n)\).

Logarithms are not just theoretical constructs; they appear in various real-world contexts like calculating compound interest and understanding sound intensity. Learning their properties makes tackling exponential equations much simpler.

Natural Logarithm

The natural logarithm, often denoted as \(\ln\), is a special logarithm where the base is the mathematical constant e (approximately 2.71828).

You often encounter natural logarithms in calculus and complex equations due to their unique properties. They are particularly useful in solving equations that involve exponential functions with base e.

Some important aspects of natural logarithms include:

• The expression \(\ln(x)\) solves for the exponent \(y\) in the equation \(e^y = x\).
• Natural logs have simple rules, such as \(\ln(e^y) = y\) and \(e^{\ln(x)} = x\).
• Natural logarithms are used in various scientific disciplines, such as physics and economics, making them integral to higher learning.

Understanding natural logarithms is essential for solving complex logarithmic equations, as seen in the sample problem where we apply \(\ln\) to both sides to simplify.

Solving Equations

Solving equations, particularly exponential equations, often involves isolating the variable of interest. The goal is to express the variable explicitly, allowing us to find its value.

Here's how to approach solving exponential equations with logarithms:

• First, transform the equation such that the exponential expression can be isolated. This often involves basic algebraic manipulation, such as division or multiplication.
• Once the base-form of the exponential is clear, apply logarithms to both sides to allow the exponent to drop down as a multiplier, simplifying the equation.
• Following simplification, further solve the equation for the unknown variable by isolating it through additional arithmetic operations such as subtraction or division.

In our solved example, we used these techniques to achieve \(x = \ln(8) - 2\), after initially transforming and then applying the natural logarithm to both sides. This systematic approach can be applied to various types of equations, underscoring the versatility and importance of logarithms in mathematics.