Problem 10 - Applied Calculus: For Business, Economics, and the Social and Life Sciences

Question

\(5^{x}=e^{3}\)

Step-by-Step Solution

Answer

x = \frac{3}{\text{ln}(5)}

Step 1: Understand the Problem

We need to solve for the variable \(x\) in the equation \(5^{x}=e^{3}\).

Step 2: Take Natural Logarithms on Both Sides

Apply the natural logarithm on both sides to get \(\text{ln}(5^{x}) = \text{ln}(e^{3})\).

Step 3: Use Logarithm Properties

Using the logarithm property \(\text{ln}(a^b) = b \text{ln}(a)\), we can rewrite the equation as \(x \text{ln}(5) = 3\).

Step 4: Solve for x

Isolate \(x\) by dividing both sides by \(\text{ln}(5)\): \(x = \frac{3}{\text{ln}(5)}\).

Key Concepts

natural logarithms

A natural logarithm, denoted as \(\text{ln}(x)\), is a type of logarithm that uses the mathematical constant 'e' (approximately 2.718) as its base. Natural logarithms are used widely in many areas of mathematics and science. In the context of our exercise, transforming the original exponential equation \(5^x = e^3\) into a more solvable form involves taking the natural logarithm of both sides. This process helps us unlock the exponent and directly deal with the variable \(x\). Remember, natural logarithms have unique properties that make them an excellent choice for solving exponential equations.

logarithm properties

To solve exponential equations like \(5^x = e^3\), we utilize properties of logarithms to simplify expressions. One of the most important properties is \(\text{ln}(a^b) = b \text{ln}(a)\). This allows us to move the exponent in front of the logarithm, making the equation linear and easier to solve. For our specific equation, applying this property transforms \(\text{ln}(5^x)\) into \(x \text{ln}(5)\). This step is crucial because it turns a complex exponentiation problem into a simpler multiplication problem, enabling us to efficiently proceed with solving for \(x\). These logarithm properties are essential tools in algebra and higher mathematics.

isolating variables

Isolating variables is a fundamental technique in algebra for solving equations. Once we've applied the logarithm properties, our equation \(x \text{ln}(5) = 3\) is set up for us to isolate \(\text{x}\). We do this by dividing both sides of the equation by \(\text{ln}(5)\). This isolates \(x\) and leaves us with the final solution: \(\text{x} = \frac{3}{\text{ln}(5)}\).
To recap:

• We simplified the original problem using natural logarithms.
• We used the property \(\text{ln}(a^b) = b \text{ln}(a)\).
• We isolated the variable by performing algebraic operations.

This process is instrumental in solving exponential equations and similar mathematical problems.