Problem 49 - Contemporary Precalculus

Question

Do the graphs of \(f(x)=\log x^{2}\) and \(g(x)=2 \log x\) appear to be the same? How do they differ?

Step-by-Step Solution

Answer

Explain why or why not. Answer: Yes, the graphs of \(f(x)=\log x^2\) and \(g(x)=2\log x\) are the same. They are equivalent functions as shown by setting them equal to each other and simplifying, and they both have the same domain of all positive real numbers \((0, +\infty)\).

Step 1: Setting Equations Equal

To find the points where the graphs intersect or if they are the same, we set \(f(x)=g(x)\): \(\log x^2 = 2\log x\)

Step 2: Solving for x

To solve for \(x\), we can use the properties of logarithms: \(\frac{\log x^2}{\log x} = \frac{2\log x}{\log x}\) Both sides can be simplified by canceling out the common logarithm terms: \(\frac{2\log x}{\log x} = 2\) For the left side, we can use the power rule for logarithms: \(\log x^2 = 2\log x\) Now, we have: \(2\log x = 2\log x\) Since both sides of the equation are equal, all values of \(x\) that are within the domain of the two functions will satisfy the equation.

Step 3: Analyzing the Domain of Functions

The domain of a logarithmic function is the set of all positive numbers. Therefore, the domain for both \(f(x)\) and \(g(x)\) is all real numbers greater than \(0\). Domain of \(f(x)\): \[x \in (0, +\infty)\] Domain of \(g(x)\): \[x \in (0, +\infty)\]

Step 4: Comparing the Functions and Making Conclusions

The equations \(f(x)=\log x^2\) and \(g(x)=2\log x\) have been shown to be equivalent through simplification. Additionally, they both have the same domain \((0, +\infty)\), which means they cover the same range of \(x\) values for their graphs. Therefore, the graphs of the functions \(f(x)=\log x^2\) and \(g(x)=2\log x\) appear to be the same, as they are equivalent functions and cover the same domain.

Key Concepts

Properties of Logarithms

Understanding the properties of logarithms is critical for solving complex logarithmic expressions and equations effectively. One key property is the 'power rule', which states that \( \log(a^n) = n\log(a) \), where \( a \) is the base of the logarithm and \( n \) is the exponent. This property is essential in simplifying logarithmic expressions like \( \log(x^2) \) to \( 2\log(x) \).

Other pivotal properties include the 'product rule' and the 'quotient rule'. The product rule tells us that \( \log(ab) = \log(a) + \log(b) \) and the quotient rule that \( \log(\frac{a}{b}) = \log(a) - \log(b) \), which are used to break down complex logarithmic statements into more manageable parts. These properties are fundamental when reshaping equations to isolate the variable, thereby streamlining the process of finding solutions to logarithmic equations.

Solving Logarithmic Equations

To solve logarithmic equations, it can be helpful to employ the properties of logarithms. For instance, when faced with an equation like \( \log(x^2) = 2\log(x) \), we can use the power rule to simplify both sides. Additionally, converting a logarithmic equation to exponential form is a common technique which involves using the definition of a logarithm, where \( \log_b(a) = c \) becomes \( b^c = a \).

However, it's critical to check the solutions against the domain of logarithmic functions, as logs are only defined for positive numbers. If solutions do not fit within the permitted domain, they must be dismissed as extraneous. Applying these concepts thoughtfully ensures that students can tackle various logarithmic equations with confidence.

Domain of Logarithmic Functions

The domain of a logarithmic function like \( f(x) = \log(x) \) is strictly the set of all positive real numbers. In other words, \( x \) must be greater than zero. This restriction stems from the fact that a logarithm represents the power to which a base must be raised to obtain some number, and raising a positive base to any real power cannot yield a negative number or zero.

Therefore, when we analyze logarithmic functions such as \( f(x) = \log(x^2) \) and \( g(x) = 2\log(x) \) from the exercise, we see that, despite their different appearances, they share the same domain, which is \( (0, +\infty) \). This commonality is vital for understanding the range of valid inputs for these functions and has significant implications when interpreting their graphs, solving equations, or applying them in real-world contexts.