Problem 99 - College Algebra Essentials

Question

Determine whether each equation is true or false. Where possible, show work to support your conclusion. If the statement is false, make the necessary change(s) to produce a true statement. \(\log _{6}\left(\frac{x-1}{x^{2}+4}\right)=\log _{6}(x-1)-\log _{6}\left(x^{2}+4\right)\)

Step-by-Step Solution

Answer

The given equation \( \log _{6}\left(\frac{x-1}{x^{2}+4}\right)=\log _{6}(x-1)-\log _{6}\left(x^{2}+4\right) \) is true as it correctly applies the logarithmic law that the log of a division of two numbers is equal to the difference of their individual logs.

Step 1: Understand the log law

Logarithms have a property that the log of a division of two numbers is equal to the difference of their individual logs. This means that, generally, if you have \( \log_{b} \left( \frac{a}{c} \right) \), you can rewrite it as \( \log_{b}(a) - \log_{b}(c) \).

Step 2: Apply the log law to the given equation

Looking at the equation \( \log _{6}\left(\frac{x-1}{x^{2}+4}\right)=\log _{6}(x-1)-\log _{6}\left(x^{2}+4\right) \), you see that the left side is a log of a division, which, according to the log law, can be taken as the difference of the individual logs. This is exactly what is seen on the right side of the equation.

Step 3: Conclusion

Since both sides of the equation reflect the same operation and the logarithmic law was rightly implemented in the right hand side, we can conclude that the given equation is true.

Key Concepts

Logarithmic Equations

Solving logarithmic equations is all about understanding how to manipulate expressions using logarithms. These equations contain unknowns within a logarithmic function. For instance, when you have an equation like \(\log_{b}(x) = y\), it implies that \(b^y = x\). But more complex examples involve properties of logarithms to simplify the equation, allowing us to isolate the variable and solve.

Consider an equation \( \log_{a}(x) + \log_{a}(y) = \log_{a}(z) \). This equation suggests that \(x \cdot y = z\), by using the property that \( \log_{a}(x \cdot y) \equiv \log_{a}(x) + \log_{a}(y) \). So one strategy in solving logarithmic equations is transforming them into an equivalent exponential form, simplifying, and then finding the values of the unknowns. It's essential to verify our solutions, as sometimes we might introduce extraneous solutions which don't satisfy the original equation.

Properties of Logarithms

Discovering the properties of logarithms can feel like unlocking new levels in a puzzle game. Each property provides a new tool for simplifying complex logarithmic expressions. Some basic properties include:

• \textbf{Product Rule:} \( \log_{b}(xy) \equiv \log_{b}(x) + \log_{b}(y) \), which shows how to split a logarithm of a multiplication into a sum of logarithms.
• \textbf{Quotient Rule:} \( \log_{b}(\frac{x}{y}) \equiv \log_{b}(x) - \log_{b}(y) \), as seen in the original exercise, allows us to express the log of a division as the difference of logs.
• \textbf{Power Rule:} \( \log_{b}(x^{n}) \equiv n \cdot \log_{b}(x) \), this tells us how to handle a logarithm of a power of a number.
• \textbf{Change of Base Formula:} For any positive numbers \(a\) and \(b\), neither of which equals \(1\), and any positive number \(x\), \( \log_{a}(x) \equiv \frac{\log_{b}(x)}{\log_{b}(a)} \).

Each property is a strategy to manage complexity - turning tough logarithmic expressions into a series of simpler operations.

Verifying Logarithmic Identities

Verifying logarithmic identities is akin to doing detective work - we scrutinize every part to confirm its truthfulness. An identity is a mathematical statement that is always true, regardless of the values we insert into it, given they are within the domain of the log functions.

For instance, one might encounter the challenge to prove that \( \log_{b}(x) + \log_{b}(y) = \log_{b}(xy) \). To verify, we can convert the logarithmic terms to their exponential forms and check if it holds true for all permissible values of \(x\) and \(y\).

If the identity is not true, as could be the case when dealing with more complex or incorrectly assumed identities, one would have to manipulate the equation using the properties of logarithms to either prove it as an identity or to find the necessary modifications to make it true. This methodological approach to confirming identities ensures not just mechanical practice but also a deeper understanding of logarithmic functions and their inherent truths.