Problem 63 - Algebra 2

Question

Solve each equation. $$ 2 \log _{3} x-\log _{3} 4=2 $$

Step-by-Step Solution

Answer

The solution to the equation is x = 6.

Step 1: Combine Logarithms

Use the properties of logarithms to combine the logarithms on the left side of the equation. This can be achieved by applying the rule \(\log_b(a) - \log_b(b) = \log_b(a/b)\). Thus, the equation simplifies to \(\log_{3}(\frac{x^2}{4}) = 2\)

Step 2: Convert the Logarithm to an Exponential Form

Solve the equation by converting it from logarithmic form to exponential form. These forms are related as follows: if \(\log_b(a) =c\), then \(b^c = a\). Thus, the equation becomes \(3^2 = \frac{x^2}{4}\) or \(9 = \frac{x^2}{4}\)

Step 3: Solve for x

Isolate x by multiplying both sides of the equation by 4. This gives \(x^2 = 36\). The positive square root of 36 is 6, so the possible value of x is 6. Logarithms do not have negative values, so -6 cannot be a solution.

Key Concepts

Properties of Logarithms

The properties of logarithms simplify complex logarithmic expressions. They help you combine or break apart logarithms for easier problem-solving. A key property used in solving equations is the quotient property. It states that:

• \(\log_b(a) - \log_b(b) = \log_b\left(\frac{a}{b}\right)\)

In our exercise, this property allows us to combine \(2 \log_3 x - \log_3 4\) into a single logarithm: \[\log_3\left(\frac{x^2}{4}\right)\] Using these properties can greatly simplify solving equations, making them powerful tools in algebra.

Exponential Form

The exponential form is a way of expressing logarithms that can make solving equations much easier. If you have an equation in logarithmic form \(\log_b(a) = c\), you can convert it to exponential form as \(b^c = a\). This transformation is crucial because it allows you to work directly with powers and roots, simplifying the process. In the exercise, the equation \(\log_3\left(\frac{x^2}{4}\right) = 2\) translates, using exponential form, to: \[3^2 = \frac{x^2}{4}\] This conversion can make complex logarithmic equations much easier to solve, making it a valuable step in the process.

Solving Equations

Solving equations often involves isolating the variable you want to find. Once in exponential form, you can solve for \(x\) by simplifying each side of the equation. For instance:

• Start with \(3^2 = \frac{x^2}{4}\), which simplifies to \(9 = \frac{x^2}{4}\).
• Multiply both sides by 4 to isolate \(x^2\), giving \(x^2 = 36\).

This process involves basic algebraic skills like multiplication and understanding the properties of equality, enabling you to find the value of \(x\) systematically.

Square Roots

The square root is the inverse operation of squaring a number. When you solve \(x^2 = 36\), you find \(x\) by taking the square root of both sides.

• The square root of 36 is 6, so \(x\) can be 6.

However, square roots technically offer both positive and negative solutions. Yet, in this context, since \(x\) was part of a logarithm expression, it must remain positive, as logarithms of negative numbers aren't defined. Understanding when to use the positive or negative root is crucial, especially when dealing with real-world problems.