Problem 1 - Algebra for College Students

Question

Solve each exponential equation and express approximate solutions to the nearest hundredth. $$ 3^{x}=13 $$

Step-by-Step Solution

Answer

x ≈ 2.34

Step 1: Understand the Equation

We start with the equation \(3^{x} = 13\). This is an exponential equation where we need to find the value of \(x\) that makes the left side equal to 13.

Step 2: Apply the Logarithm

To solve for \(x\), we take the natural logarithm (ln) of both sides of the equation. This gives us: \( \ln(3^{x}) = \ln(13) \). Using the power rule for logarithms, this becomes \( x \cdot \ln(3) = \ln(13) \).

Step 3: Solve for x

Now, isolate \(x\) by dividing both sides of the equation by \(\ln(3)\): \( x = \frac{\ln(13)}{\ln(3)} \).

Step 4: Calculate x

Calculate the value using a calculator: \( \ln(13) \approx 2.5649 \) and \( \ln(3) \approx 1.0986 \). Thus, \( x = \frac{2.5649}{1.0986} \approx 2.34 \).

Key Concepts

Natural Logarithms

Natural logarithms are a type of logarithm where the base is the mathematical constant \(e\), approximately equal to 2.71828. They are denoted as \(\ln\), which stands for 'logarithm naturalis.' Natural logarithms are particularly useful in solving equations involving exponential expressions. This is because they help transform exponential terms into more manageable algebraic terms.

When solving exponential equations, one common technique is to take the natural logarithm of both sides of the equation. This helps by using logarithmic identities to simplify the exponential term. For instance, if you have \(3^x = 13\), taking the natural logarithm of both sides yields \(\ln(3^x) = \ln(13)\).

A key property of logarithms leveraged here is the power rule: \(\ln(a^b) = b \cdot \ln(a)\). This allows you to 'bring down' the exponent \(x\) as a factor, transforming the equation into \(x \cdot \ln(3) = \ln(13)\). This move is crucial in isolating the variable \(x\), making these types of equations much easier to solve.

Solving Equations

Solving equations involves finding the value of the variable that satisfies the equation. In the case of exponential equations, like \(3^x = 13\), this means determining the value of \(x\) that makes both sides equal.

The process starts with applying the natural logarithm to both sides, transforming the exponential equation into an algebraic one. Through this method, the exponential component becomes a product according to the power rule of logarithms, simplifying to \(x \cdot \ln(3) = \ln(13)\).

To solve for \(x\), divide both sides of the equation by \(\ln(3)\). This isolation of \(x\) yields \(x = \frac{\ln(13)}{\ln(3)}\).

Here is the step-by-step breakdown:

• Take the natural logarithm of both sides.
• Apply the power rule: \(x \cdot \ln(3) = \ln(13)\).
• Isolate \(x\) by dividing by \(\ln(3)\).

With these steps, you can transition from an exponential form to a simpler algebraic expression, making it easier to find the solution.

Mathematical Approximation

Mathematical approximation is the process of finding a value that is not exact but is close enough to the correct answer for practical purposes. This is particularly useful when dealing with irrational numbers or complex calculations that are challenging to compute by hand.

In our exercise, after isolating \(x\), we need to calculate \(x = \frac{\ln(13)}{\ln(3)}\) using approximate values. Calculators provide natural logarithm values, so we use approximate values: \(\ln(13) \approx 2.5649\) and \(\ln(3) \approx 1.0986\).

By dividing these values, you estimate \(x \approx 2.34\). This is an example of rounding to the nearest hundredth, a common requirement in mathematical approximation tasks.

Approximation helps in simplifying real-world problems and making calculations feasible. Precision is key, but so is practicality, especially in educational settings where exact values may not be necessary. Use approximation smartly to efficiently solve equations while maintaining an acceptable degree of accuracy.