Problem 50 - Contemporary Precalculus

Question

Do the graphs of \(h(x)=\log x^{3}\) and \(k(x)=3 \log x\) appear to be the same?

Step-by-Step Solution

Answer

Answer: Yes, the graphs of the functions \(h(x)=\log x^{3}\) and \(k(x)=3\log x\) appear the same because their simplified versions are equal.

Step 1: Rewrite the first function using logarithmic properties

Recall that \(\log a^{b}=b\log a\). Apply this property to the function \(h(x)=\log x^{3}\): \(h(x)=3\log x\) Now, the function h(x) is simplified as: \(h(x)=3\log x\)

Step 2: Rewrite the second function using logarithmic properties

The second function is already simplified: \(k(x)=3\log x\)

Step 3: Analyze the two simplified functions

We now have the following functions: \(h(x)=3\log x\) \(k(x)=3\log x\) Looking at the two functions, we can see that they are the same.

Step 4: Conclusion

Since the simplified versions of the functions h(x) and k(x) are equal, the graphs of \(h(x)=\log x^{3}\) and \(k(x)=3\log x\) will indeed appear the same.

Key Concepts

Properties of Logarithms

Logarithms are powerful mathematical tools that help simplify complex problems. A key property of logarithms is that it transforms multiplication into addition, which is immensely helpful in computations. One significant property is:

• Power Rule: \( \log(a^b) = b \log(a) \)

This means any exponent within the logarithm can be moved to the front as a multiplier. For example, when we have \( h(x) = \log(x^3) \), we can use the power rule to simplify it to \( h(x) = 3 \log(x) \).
This transformation shows us that the function can be rewritten, making it easier to compare or graph.
Using logarithmic properties like these not only helps in solving equations but also in understanding function behavior in a graphical sense.

Graphing Functions

Graphing functions is a foundational way to understand their behavior. When we graph logarithmic functions, it's helpful to know that they typically have a characteristic shape. Key points to remember include:

• The domain of a logarithmic function \( \log(x) \) is \( x > 0 \).
• The range is all real numbers.
• Logarithmic graphs usually pass through the point \((1, 0)\).

The process to graph functions like \( h(x) = 3 \log(x) \) and \( k(x) = 3 \log(x) \) involves determining key points and the overall shape of the graph.
Since both functions simplify to the same expression, their graphs will overlap completely. Each has the same intercept and rate of change because they are identical.
Understanding the graphical representation of logarithmic functions helps visualize solutions and confirm algebraic simplifications.

Function Simplification

Simplifying functions is a process where we use algebraic techniques to reduce expressions to their most basic form. This is crucial for understanding the true nature of a function and making computations easier. In the case of the functions \( h(x) = \log(x^3) \) and \( k(x) = 3 \log(x) \):

• Both can be rewritten as \( 3 \log(x) \) through logarithmic properties.

Simplification allows us to see that these functions are indeed identical. This process is not just about reducing complexity but also about revealing underlying equivalences.
A simplified function can show us overlaps in solutions, give insights into possible symmetries, and provide a clearer path to graphing.
It's a fundamental skill in algebra and calculus that builds confidence in solving more complex mathematical problems.