Problem 60 - College Algebra with Modeling and Visualization

Question

Complete the following. A.Use a table of \(f(x)\) and \(g(x)\) to determine whether \(f(x)=g(x) B.If possible, use properties of logarithms to show that \)f(x)=g(x)$ $$ f(x)=\log x^{2}+\log x^{3}, \quad g(x)=5 \log x $$

Step-by-Step Solution

Answer

Yes, \( f(x) = g(x) \).

Step 1: Simplify f(x) using properties of logarithms

Given \[ f(x) = \log(x^{2}) + \log(x^{3}) \]Use the property of logarithms that states \( \log(a) + \log(b) = \log(ab) \):\[ f(x) = \log(x^{2} \cdot x^{3}) \]Thus, \( f(x) \) simplifies to \( \log(x^{5}) \).

Step 2: Simplify f(x) further

Use the power rule of logarithms, which states \( \log(a^b) = b \cdot \log(a) \):\[ f(x) = \log(x^{5}) = 5 \cdot \log(x) \]

Step 3: Compare f(x) and g(x)

We have:\[ f(x) = 5 \cdot \log(x) \]And given \( g(x) = 5 \cdot \log(x) \).Since both functions are expressed as \( 5 \cdot \log(x) \), we can conclude: \[ f(x) = g(x) \].

Key Concepts

Properties of Logarithms

The properties of logarithms simplify our work with logarithmic expressions immensely. Two important properties come into play here: the product property and the power rule.

The **product property** states that if you have a sum of two logarithms with the same base, you can combine them into a single logarithm of the product of their arguments. Mathematically, this becomes \( \log(a) + \log(b) = \log(ab) \). This is seen in the exercise, where \( \log(x^2) + \log(x^3) \) is combined into \( \log(x^2 \cdot x^3) \).

Once combined, we can further simplify the expression using the **power rule**, which lets us "bring the exponent out in front," greatly simplifying expressions. This principle is expressed as \( \log(a^b) = b \cdot \log(a) \). In our example, \( \log(x^5) \) becomes \( 5 \cdot \log(x) \), allowing an easier comparison with other expressions in the problem.

Logarithmic Equations

Logarithmic equations often involve expressions set equal to each other, requiring simplification for solving. Here, we have two functions, \( f(x) \) and \( g(x) \), which look different but can be equivalent through simplification.

The task is to demonstrate \( f(x) = g(x) \). After using the properties of logarithms to simplify \( f(x) \) as we have done, we can express it in a form that can be directly compared to \( g(x) = 5 \cdot \log(x) \).

Here's how the process unfolds:

• Simplify each side using logarithmic properties so they appear similar.
• Compare both sides after simplification. If they match, the equation holds true.

This reduces the complexity of solving logarithmic equations, making it easier to identify equivalent equations.

Mathematical Proofs

Proving that two expressions are equivalent can be a very satisfying endeavor, and mathematics provides structured methods for validation. In our example, the goal is to prove that \( f(x) = g(x) \) through logical steps.

Proofs using logarithms rely on careful application of algebraic rules. Here's the approach:

• Apply known \( \text{logarithmic identities} \), such as the product rule and power rule, to simplify each expression.
• Reformulate both expressions in the same form, ready for comparison.
• Once both expressions are identical, the proof is complete.

Mathematical proofs provide a step-by-step logic flow, ensuring that equivalent functions or equations are unquestionably verified, much like verifying \( f(x) = g(x) = 5 \cdot \log(x) \) in this problem.