Problem 21 - Precalculus

Question

Solve the exponential equations. Make sure to isolate the base to a power first. Round our answers to three decimal places. $$2\left(3^{x}\right)-11=9$$

Step-by-Step Solution

Answer

\( x \approx 2.097 \)

Step 1: Isolate the Exponential Term

Start with the equation \( 2(3^x) - 11 = 9 \). The goal is to isolate \( 3^x \). First, add 11 to both sides of the equation to get \( 2(3^x) = 20 \).

Step 2: Simplify the Equation

Divide both sides of the equation by 2 to solve for \( 3^x \), giving \( 3^x = 10 \). Now, our equation is in the form where the exponential base is isolated.

Step 3: Solve for x Using Logarithms

Take the logarithm of both sides to solve for \( x \). Therefore, \( \log(3^x) = \log(10) \). Using the properties of logarithms, this becomes \( x \cdot \log(3) = \log(10) \).

Step 4: Calculate x

Solve for \( x \) by dividing both sides by \( \log(3) \), so \( x = \frac{\log(10)}{\log(3)} \). Using a calculator, find the values: \( \log(10) \approx 1 \) and \( \log(3) \approx 0.4771 \). Therefore, \( x \approx \frac{1}{0.4771} \approx 2.097 \).

Step 5: Round to Three Decimal Places

The calculated value of \( x \) was already rounded to three decimal places in Step 4, resulting in \( x \approx 2.097 \).

Key Concepts

Logarithms

Logarithms are incredibly useful mathematical tools, particularly when you're dealing with exponential equations, as they allow you to solve for unknown exponents. Think of a logarithm as the inverse operation to exponentiation. If you have a number factored in terms of a base raised to an exponent, you can resolve back to the exponent using logarithms.
For instance, when working with the equation where the exponential term is isolated, converting it involves taking the logarithm of both sides. In the problem above, we faced the expression \( 3^x = 10 \). By applying the logarithm, the expression becomes \( \log(3^x) = \log(10) \).
This step doesn't alter the equation's truth; instead, it merely transforms it into a form where calculating the exponent is straightforward. The logarithmic property \( \log(3^x) = x \cdot \log(3) \) makes it easier to isolate \( x \), which is our unknown.

• Logarithms effectively "bring down" the exponent.
• Helps in solving equations where the variable is in the exponent form.

Through the use of a scientific calculator, you can then compute these values easily to find your desired variable. In our case, this led us to solve \( x \approx \frac{1}{0.4771} \), thanks to the properties of logarithms.

Isolating Variables

Isolating variables is a fundamental skill in algebra and is crucial for solving equations. When you're dealing with equations, especially ones involving exponential terms, the first step is often to "isolate" the variable expression. By doing this, you set up the equation in a way that allows you to solve for the variable.
In our original problem, \( 2(3^x) - 11 = 9 \), isolating \( 3^x \) required moving all other terms to the opposite side of the equation. This entailed:

• Adding 11 to both sides to eliminate the \(-11\) from the left side.
• Dividing both sides by 2 to remove the coefficient of the exponential term.

After these algebraic manipulations, we got \( 3^x = 10 \), successfully isolating the exponential expression. This is key because it allowed us to apply the logarithm in the next step.
Isolating variables simplifies the entire process of solving equations,

• Makes complex problems more manageable by reducing them to simpler steps.
• Creates a clear path for using other mathematical operations such as logarithms or division.

Exponential Functions

Exponential functions frequently appear in mathematics, notably in growth processes and decay patterns. Such functions have the general form of \( f(x) = a^x \), where \( a \) is a positive constant. The rate of change in these functions is often dramatic, increasing or decreasing faster than linear functions as \( x \) becomes larger or more negative.
In the equation \( 2(3^x) - 11 = 9 \), the term \( 3^x \) signifies an exponential function because it represents a constant base \( 3 \) raised to a variable power \( x \). Understanding this helps us recognize how to manage the function through algebraic manipulations like isolating or applying logarithms.
Challenges in solving exponential equations arise from the variable being in the exponent. Hence, turning to logarithms becomes a necessary strategy. Recognizing and manipulating exponential functions:

• Requires understanding how the base and exponent relate.
• Explains diverse phenomena like population growth, radioactive decay, and more.
• Emphasizes the importance of knowing the properties of exponents and their inverses—logarithms.

In practice, mastering these concepts equips you with the essential tools to solve real-world problems involving exponential relationships.