Problem 17 - Precalculus

Question

Apply the properties of logarithms to simplify each expression. Do not use a calculator. $$7^{-2 \log _{7} 3}$$

Step-by-Step Solution

Answer

The simplified expression is \(\frac{1}{9}\).

Step 1: Understand the Given Expression

The expression given is \(7^{-2 \log_{7} 3}\). It has a base of 7 raised to the power of \(-2 \log_{7} 3\). Our aim is to simplify this expression using properties of logarithms.

Step 2: Apply the Logarithm Power Rule

Recall the logarithm power rule: if you have \(a^{\log_{a} b} = b\). We need to rewrite the exponent part of the expression in a similar form. Notice here we can manipulate the exponent so it takes this form.

Step 3: Simplify the Exponent

There is a property of exponents that allows us to rewrite \(-2 \log_{7} 3\) as \(\log_{7}(3^{-2})\). This uses the property that \(-c \log_{b} a = \log_{b}(a^{-c})\).

Step 4: Apply the Inverse Logarithmic Property

Using the inverse of the logarithmic exponential property, simplify the base and the rewritten exponent: \(7^{\log_{7}(3^{-2})} = 3^{-2}\). This is because \(a^{\log_{a}(b)} = b\), which in this case translates to \(7^{\log_{7}(3^{-2})} = 3^{-2}\).

Step 5: Compute the Simplified Expression

The expression \(3^{-2}\) can be evaluated as \(\frac{1}{3^{2}} = \frac{1}{9}\). Thus, \(7^{-2 \log_{7} 3} = \frac{1}{9}\).

Key Concepts

Properties of Logarithms

When tackling logarithmic expressions, it's essential to understand the fundamental properties of logarithms. These properties help us manipulate and simplify expressions. Here are some common properties:

• Product Rule: \( \log_{b}(mn) = \log_{b}m + \log_{b}n \). This allows us to separate logarithms with multiplied arguments into the sum of two logs.
• Quotient Rule: \( \log_{b}(\frac{m}{n}) = \log_{b}m - \log_{b}n \). This rule helps in splitting a division within the log into a difference.
• Change of Base: \( \log_{b}a = \frac{\log_{c}a}{\log_{c}b} \). This is useful for converting logs into different bases.

Understanding how these properties work allows us to break down complex logarithmic expressions into more manageable parts. In the exercise, we used the logarithm power rule to help simplify the expression.

Logarithm Power Rule

The logarithm power rule is a handy tool when simplifying expressions involving exponents with logs. The rule states that \( a^{\log_{a} b} = b \). This means that when a base is raised to a logarithm with the same base, it simplifies directly to the argument of the logarithm.
In the given problem, we aimed to express the exponent in a form that fits this rule. The expression was initially \( 7^{-2 \log_{7} 3} \). We reworked the exponent using this rule by expressing it as \( \log_{7}(3^{-2}) \). Thus, allowing us to simplify the logarithmic expression efficiently.

Inverse Log Property

The inverse log property states that when you have a situation where the base of a power and the base of a logarithm are the same, such as \( a^{\log_{a}(b)} \), it simplifies to \( b \). This property is pivotal in simplifying the expression we started with.
In our example, once we expressed the exponent as \( \log_{7}(3^{-2}) \), we could apply the inverse log property: \(7^{\log_{7}(3^{-2})} = 3^{-2}\). It's a powerful simplification tool because it cuts through the complexity to provide the answer in one simple step.
Recognizing when you can apply this property is crucial for efficiently handling logarithmic exercises.

Simplification of Expressions

When simplifying complex expressions, whether involving logarithms or not, we seek to write the expression in its simplest form. This often involves using mathematical properties strategically, as we demonstrated in the given problem. Here's how the simplification steps were applied:

• First, identify parts of the expression that align with a known property (like logarithm rules).
• Rewrite those parts to leverage these properties into a simpler form (e.g., turning the exponent \(-2 \log_{7} 3\) into \(\log_{7}(3^{-2})\)).
• Use simplification properties such as the inverse log property to further reduce the expression.
• Continue simplifying, if possible, until you reach the most reduced form, such as taking \(3^{-2}\) and expressing it as \(\frac{1}{9}\).

By breaking it down into these manageable steps, any mathematical expression becomes less intimidating and easier to solve. Simplification is the final step aiming for clarity and the most straightforward expression form.