Problem 20 - Algebra and Trigonometry with Analytic Geometry

Question

Solve the equation. $$ 3 \log _{2} x=2 \log _{2} 3 $$

Step-by-Step Solution

Answer

The solution is \( x = \sqrt[3]{9} \).

Step 1: Apply the Logarithmic Property

Start by using the property of logarithms that states if you have a constant multiplied by a logarithm, you can rewrite it as a power inside the logarithm. In this problem, rewrite both logarithms:\[ 3 \log_{2} x = \log_{2} (x^3) \quad \text{and} \quad 2 \log_{2} 3 = \log_{2} (3^2) \]

Step 2: Simplify the Equation

Now that both sides of the equation are expressed as single logarithms, the equation becomes:\[ \log_{2} (x^3) = \log_{2} (9) \]

Step 3: Use the Logarithm Equality Property

Since the logarithms have the same base, the expressions inside must be equal. Set the expressions inside the logarithms equal to each other:\[ x^3 = 9 \]

Step 4: Solve for x

To solve for \( x \), take the cube root of both sides:\[ x = \sqrt[3]{9} \]

Key Concepts

Properties of Logarithms

When solving logarithmic equations, understanding the properties of logarithms is key. These properties allow us to manipulate and simplify expressions involving logarithms, making equations easier to solve. Here are a few essential properties:

• Logarithm of a Power: If you have a logarithm with a power, such as \( \log_b(x^a) \,\) you can rewrite it as \( a \log_b x \).
• Product and Quotient Rules: The logarithm of a product \( \log_b(xy) = \log_b x + \log_b y\), and the logarithm of a quotient \( \log_b(x/y) = \log_b x - \log_b y\).
• Change of Base Formula: This helps convert logarithms from one base to another, allowing flexible calculations: \( \log_b a = \frac{\log_k a}{\log_k b}\).

In our exercise, we utilized the logarithm of a power property. By rewriting \( 3 \log_{2} x \) as \( \log_{2} (x^3) \,\) and \( 2 \log_{2} 3 \) as \( \log_{2} (3^2) \,\) we transformed the equation into a simpler form.

Logarithm Equality

The concept of logarithm equality is a powerful tool for solving equations involving logarithms. It states that if two logs with the same base are equal, the expressions inside those logs must also be equal.

• For example, if \( \log_b a = \log_b c \,\) then \( a = c \,\) providing the logarithms have the same base.

In the original exercise, after rewriting both sides of the equation with the property of logarithms, the equation was simplified to \( \log_{2} (x^3) = \log_{2} (9) \,\). Because the logarithms are on the same base, we could directly set the expressions equal: \( x^3 = 9 \), leveraging the property of logarithm equality to simplify our problem.

Solving Equations Involving Logarithms

Solving equations involving logarithms requires clearly following steps, often relying on the properties of logarithms and the equality principle. Here's a basic approach:

• Step 1: Try to express the equation such that both sides become single logarithmic forms, using properties of logarithms to simplify.
• Step 2: Employ the logarithm equality to equate the expressions inside the logarithms if they share the same base.
• Step 3: Solve for the unknown variable using algebraic techniques such as powers or roots.

In the provided exercise, the final step was to solve \( x^3 = 9 \.\) We took the cube root of both sides to find the solution \( x = \sqrt[3]{9} \,\) showcasing how knowing algebra techniques along with logarithmic properties can aid in finding solutions to complex equations. Understanding each step thoroughly helps in accurately and confidently solving logarithmic equations.