Problem 8 - A Graphical Approach to College Algebra

Question

Solve each equation. $$\left(\frac{2}{3}\right)^{x}=\frac{9}{4}$$

Step-by-Step Solution

Answer

The solution is \( x \approx -1.226 \).

Step 1: Take the logarithm of both sides

Start by taking the logarithm of both sides of the equation: \( \log \left( \left( \frac{2}{3} \right)^x \right) = \log \left( \frac{9}{4} \right) \). This helps in bringing down the exponent for easier manipulation.

Step 2: Apply the power rule of logarithms

Utilize the power rule of logarithms, which states \( \log(a^b) = b \cdot \log(a) \). Thus, rewrite the equation as \( x \cdot \log \left( \frac{2}{3} \right) = \log \left( \frac{9}{4} \right) \).

Step 3: Solve for x

Isolate \( x \) by dividing both sides by \( \log \left( \frac{2}{3} \right) \). The equation becomes: \( x = \frac{\log \left( \frac{9}{4} \right)}{\log \left( \frac{2}{3} \right)} \).

Step 4: Use a calculator to compute the values

Calculate the logarithmic values using a calculator. First, compute \( \log \left( \frac{9}{4} \right) \) and then \( \log \left( \frac{2}{3} \right) \). Divide the results to find \( x \).

Key Concepts

Logarithms

Logarithms are incredibly useful in mathematics, particularly when handling exponential equations like the one in our exercise. Logarithms can be thought of as the "opposite" of exponentiation. For example, if you know that \( b^y = x \), then the logarithm helps us express \( y \) as \( \log_b(x) \). Essentially, it asks "what power \( y \) do we apply to base \( b \) to get \( x \)?".

In solving our equation, using logarithms allowed us to transform the equation from an exponential form, \( \left(\frac{2}{3}\right)^x\), into a more manageable form, making \( x \) much easier to isolate.

Power Rule of Logarithms

The power rule of logarithms is a handy tool when dealing with equations involving exponents. It states that \( \log(a^b) = b \cdot \log(a) \).

In simple terms, this rule allows us to take any exponent in a logarithmic expression and bring it down in front as a multiplier.

So, in our equation \( \log \left( \left( \frac{2}{3} \right)^x \right) \), the power rule helps us rewrite it as \( x \cdot \log \left( \frac{2}{3} \right) \).

• Reduces the complexity of exponential equations
• Enables easier manipulation and simplification
• Allows us to solve for the variable by straightforward algebraic methods

Solving Equations Step by Step

Solving exponential equations involves breaking the process into bite-sized steps to avoid confusion and errors. Here’s a streamlined approach:

• **Step 1: Transform using logarithms.** Apply logarithms on both sides of the equation to bring down the exponent.
• **Step 2: Simplify using the power rule of logarithms.** Convert the difficult exponential equation into a manageable linear form.
• **Step 3: Isolate the variable.** Rearrange the equation to solve for the unknown variable.
• **Step 4: Compute values.** Use a calculator to find numerical results for the logarithms and solve for the variable's value.

Taking these steps systematically makes the problem less daunting and allows one to focus on solving each part correctly. Patience and attention to detail are key!