Question

Solve each exponential equation. Express irrational solutions in exact form. $$ e^{x+3}=\pi^{x} $$

Short answer

\( x = \frac{-3}{1 - \text{ln}(\pi)} \)

Step-by-step solution

- Take natural logarithms on both sides

To solve the equation, take the natural logarithm (ln) on both sides: \( \text{ln}(e^{x+3}) = \text{ln}(\pi^x) \)

- Use the properties of logarithms

Using the properties of logarithms, simplify both sides. For the left side, use \( \text{ln}(e^a) = a \). For the right side, use \( \text{ln}(a^b) = b \text{ln}(a) \): \( x + 3 = x \text{ln}(\pi) \)

- Solve for x

Isolate \( x \) on one side of the equation: \( x + 3 = x \text{ln}(\pi) \) Subtract \( x \text{ln}(\pi) \) from both sides: \( x + 3 - x \text{ln}(\pi) = 0 \) Factor out \( x \) on the left side: \( x(1 - \text{ln}(\pi)) = -3 \) Divide both sides by \( 1 - \text{ln}(\pi) \): \( x = \frac{-3}{1 - \text{ln}(\pi)} \)

Key concepts

Natural Logarithms

Natural logarithms, often abbreviated as **ln**, are logarithms with a base of **e**. The number **e** is an irrational constant approximately equal to 2.71828. Natural logarithms are especially useful when dealing with equations involving exponential functions, particularly those with base **e**.

In the given problem, we start by taking the natural logarithm on both sides of the equation. This step simplifies the exponential expressions, making the equation easier to handle.

Example: For the equation \(e^{x+3} = \text{ln}(\text{int}\big(\frac{245}{103})\)), we apply **ln** to both sides:

\[\ln(e^{x+3})=\ln(\pi^x)\]

The purpose of using **ln** is to make use of logarithmic properties that can help simplify the expressions.

Properties of Logarithms

Logarithms have several useful properties that make solving equations more manageable. Here are some essential properties used in our problem:

\(\text{ln}(e^a) = a\)

\(\text{ln}(a^b) = b \text{ln}(a)\)

Let's apply these properties to our equation:

After taking the natural logarithms, we have:

\[\ln(e^{x+3}) = \ln(\pi^x)\]

Using the property \(\text{ln}(e^a) = a\), the left side simplifies to \(x + 3\).

For the right side, using the property \(\text{ln}(a^b) = b \text{ln}(a)\), it becomes \(x \text{ln}(\pi)\).

Therefore, the simplified form of the equation is:

\[ x + 3 = x \text{ln}(\pi)\]

The application of these properties transforms a complex exponential equation into a much simpler linear form.

Isolating Variables

Isolating the variable is a crucial step when solving equations. In this problem, our goal is to isolate **x**.

We start with the simplified logarithmic equation:

\[x + 3 = x \text{ln}(\pi)\]

To isolate **x**, first, rearrange the terms by subtracting **x ln(π)** from both sides:

\[x + 3 - x \text{ln}(\pi) = 0\]

Next, factor **x** out from the left side:

\[x(1 - \text{ln}(\pi)) + 3 =0\]

Once factored, solve for **x** by dividing both sides by \(1 - \text{ln}(\text{int}\big(\frac{245}{103})\)):

\[ x=\frac{-3}{1- \text{ln}(\pi)}\]

This isolates **x** and provides the exact solution.

The principles of isolating variables and algebraic manipulation are fundamental tools for solving more complex mathematical problems.