Problem 18 - Precalculus

Question

Use the properties of logarithms to write the expression as a single logarithm. $$ \log _{3}(x)-2 \log _{3}(y) $$

Step-by-Step Solution

Answer

The expression simplifies to \(\log_3(\frac{x}{y^2})\).

Step 1: Identify the Properties of Logarithms

To combine the given expression into a single log, we will use two key properties: the Product Rule \( \log_b(m) + \log_b(n) = \log_b(m \cdot n) \) and the Quotient Rule \( \log_b(m) - \log_b(n) = \log_b(\frac{m}{n}) \). Additionally, the Power Rule \( a \log_b(m) = \log_b(m^a) \) will help simplify terms.

Step 2: Apply the Power Rule First

In the expression \( -2\log_3(y) \), apply the power rule to move the coefficient \(-2\) as an exponent of \(y\). This transforms \(-2\log_3(y)\) to \(\log_3(y^{-2})\). Hence, the expression becomes \(\log_3(x) + \log_3(y^{-2})\).

Step 3: Use the Product Rule to Combine Logarithms

Now use the product rule: \(\log_b(m) + \log_b(n) = \log_b(m \cdot n)\). Apply it to combine \(\log_3(x)\) and \(\log_3(y^{-2})\) into a single logarithm. This results in \(\log_3(x \cdot y^{-2})\).

Step 4: Simplify the Expression

Write the expression \(x \cdot y^{-2}\) in a simplified form as \(\frac{x}{y^2}\) using the property that dividing by exponentiation is the same as using a negative exponent. This gives the final expression \(\log_3(\frac{x}{y^2})\).

Key Concepts

Properties of Logarithms

Logarithms have several properties that allow us to manipulate and transform expressions. Here are some of the most important ones:

• Product Rule: This states that the logarithm of a product is the sum of the logarithms of its factors. Mathematically, it is expressed as \( \log_b(m) + \log_b(n) = \log_b(m \cdot n) \).
• Quotient Rule: This tells us that the logarithm of a quotient is the difference between the logarithms of the numerator and the denominator: \( \log_b(m) - \log_b(n) = \log_b(\frac{m}{n}) \).
• Power Rule: If you have a logarithm of a number raised to a power, you can bring the power out in front as a factor: \( a \log_b(m) = \log_b(m^a) \).

These properties are extremely useful in simplifying logarithmic expressions, making them easier to work with.

Product Rule

The Product Rule for logarithms is a powerful tool for combining multiple logarithmic terms into a single logarithm. To apply it, remember that when you add two logarithms with the same base, it is equivalent to taking the logarithm of the product of the two arguments. For example:

• If you have \( \log_3(x) + \log_3(y^{-2}) \), you can combine them using the product rule: \( \log_3(x \cdot y^{-2}) \).

This reduces what might initially look like a complex expression into something much simpler. This is just one instance where the product rule is applied to streamline logarithmic expressions efficiently.

Power Rule

The Power Rule is another indispensable property. This rule allows you to simplify expressions by converting multiplication into exponentiation. It states: \( a \log_b(m) = \log_b(m^a) \). This means that if you have a coefficient multiplied with a logarithm, you can move that coefficient as an exponent of the logarithmic argument. Here's how it works:

• Consider the expression \(-2\log_3(y)\). Applying the power rule transforms it to \(\log_3(y^{-2})\).

The power rule is particularly helpful for simplifying complex terms within a logarithmic expression. By using this property, your algebra becomes more straightforward and easy to solve.

Simplifying Logarithms

Simplifying logarithms is all about applying the right properties to make expressions as straightforward as possible. The main goal is to reduce multi-term logarithmic expressions to single logarithms, if possible. Here's the step-by-step process for our example:

• First, apply the Power Rule: This transforms \(-2\log_3(y)\) into \(\log_3(y^{-2})\).
• Then, use the Product Rule: Combine \(\log_3(x)\) and \(\log_3(y^{-2})\) into a single logarithm: \(\log_3(x \cdot y^{-2})\).
• Finally, simplify the result: Use exponent rules to express \(x \cdot y^{-2}\) as \(\frac{x}{y^2}\), getting \(\log_3(\frac{x}{y^2})\).

Each of these steps uses a specific property of logarithms to turn a complex expression into a simpler and more elegant form.