Problem 29 - Precalculus with Calculus Previews

Question

Solve the given logarithmic equation. $$ \log _{3} 81^{x}-\log _{3} 3^{2 x}=3 $$

Step-by-Step Solution

Answer

The solution is \(x = \frac{3}{2}\).

Step 1: Apply the Power Rule

The power rule of logarithms states that \(\log_b(a^n) = n \cdot \log_b(a).\) Applying this rule to each logarithm in the equation, we have: \[\log_3(81^x) = x \cdot \log_3(81)\] \[\log_3(3^{2x}) = 2x \cdot \log_3(3).\]

Step 2: Simplify the Logarithms

Now, we will evaluate the logarithms. Note that \(81 = 3^4\), so \(\log_3(81) = \log_3(3^4) = 4\cdot\log_3(3) = 4.\) Therefore, \(x \cdot \log_3(81) = x \cdot 4 = 4x.\) The other logarithm simplifies to: \[2x \cdot \log_3(3) = 2x \cdot 1 = 2x.\]

Step 3: Substitute Simplified Logarithms into Equation

Replace the original logarithms with their simplified expressions:\[4x - 2x = 3.\] This is the result after substituting \(4x\) and \(2x\) in place of the logarithmic terms.

Step 4: Solve the Linear Equation

You are left with the simple linear equation: \[2x = 3.\] Divide both sides by \(2\) to solve for \(x\): \[x = \frac{3}{2}.\]

Step 5: Verify the Solution

To ensure our solution is correct, we can substitute \(x = \frac{3}{2}\) back into the original equation and verify that both sides equal 3. The substitution verifies without contradiction, confirming that \(x = \frac{3}{2}\) is correct.

Key Concepts

Logarithm Power Rule

The Logarithm Power Rule is a fundamental concept useful in simplifying expressions in logarithmic equations. This rule can turn the logarithm of an exponential expression into a much simpler, more workable form. The power rule states:

• \(\log_b(a^n) = n \cdot \log_b(a)\)

In our specific exercise, consider the expression \(\log_3(81^x)\). By applying the power rule, we rewrite it as \(x \cdot \log_3(81)\). The same approach is applied to \(\log_3(3^{2x})\), simplifying it to \(2x \cdot \log_3(3)\).
These conversions are helpful because they transform potentially complicated functions into linear terms that are much easier to handle. This step is crucial in solving the equation.

Solving Linear Equations

After applying the Logarithm Power Rule, we often encounter linear equations. Solving these is a key step in finding the solution to our logarithmic equation.
For example, once we simplified our problem using the power rule, we ended up with:

• \(4x - 2x = 3\)

This equation is straightforward to solve. Begin by combining like terms on the left-hand side:

• \(2x = 3\)

Next, divide each side of the equation by 2, which isolates \(x\) on one side:

• \(x = \frac{3}{2}\)

Linear equations typically have a unique solution, which in this case is \(x = \frac{3}{2}\). Understanding these steps is crucial to mastering algebraic techniques used in more complex problems.

Verification of Solutions

Verification is the process of confirming the accuracy of your solution by substituting it back into the original equation. This step is crucial because it ensures that the answer satisfies the initial conditions provided.
In our exercise, after solving for \(x\) and obtaining \(x = \frac{3}{2}\), we substitute this value back into the original logarithmic equation:

• \(\log_3(81^{3/2}) - \log_3(3^{3}) = 3\)

Then, simplify each term using the logarithm rules:

• \(\frac{3}{2} \cdot \log_3(81) - 3 \cdot \log_3(3)\)

Since \(\log_3(81) = 4\) and \(\log_3(3) = 1\), we get:

• \(\frac{3}{2} \times 4 - 3 \times 1 = 3\)
• \(6 - 3 = 3\)

The left and right sides of the original equation match perfectly, confirming that the solution \(x = \frac{3}{2}\) is indeed correct.