Logarithm Properties: True Or False?

When we dive into the world of mathematics, especially algebra and calculus, logarithm properties are fundamental tools that simplify complex expressions and equations. Understanding these properties is crucial for solving problems efficiently. Today, we're going to dissect a few statements involving logarithms to determine whether they hold true or false. This isn't just about memorizing rules; it's about understanding why these rules work and how they are derived from the basic definition of a logarithm. Let's get started and put your knowledge to the test! We'll break down each statement, explaining the relevant logarithm properties and showing the step-by-step process to arrive at the correct conclusion. By the end of this discussion, you'll have a clearer grasp of these essential mathematical concepts.

Statement 1: $\log _3\left(c d^4\right)=4 \log _3 c+4 \log _3 d$

Let's begin by examining the first statement: $\log _3\left(c d^4\right)=4 \log _3 c+4 \log _3 d$. To determine if this is true or false, we need to recall the fundamental properties of logarithms. The most relevant properties here are the product rule and the power rule for logarithms. The product rule states that $\log _b(xy) = \log _b x + \log _b y$, meaning the logarithm of a product is the sum of the logarithms of the factors. The power rule states that $\log _b(x^n) = n \log _b x$, indicating that the logarithm of a number raised to a power is the power times the logarithm of the number. Now, let's apply these rules to the left side of the given equation, $\log _3\left(c d^4\right)$.

Using the product rule, we can separate the terms $c$ and $d^4$ since they are multiplied together inside the logarithm: $\log _3\left(c d^4\right) = \log _3 c + \log _3\left(d^4\right)$. We have successfully used the product rule. Now, let's focus on the second term, $\log _3\left(d^4\right)$. Here, we can apply the power rule because $d$ is raised to the power of 4. According to the power rule, $\log _3\left(d^4\right) = 4 \log _3 d$.

Substituting this back into our expression, we get: $\log _3 c + 4 \log _3 d$. Now, let's compare this result to the right side of the original statement, which is $4 \log _3 c+4 \log _3 d$. Our derived expression is $\log _3 c + 4 \log _3 d$. Clearly, these two expressions are not the same. Specifically, the term $\log _3 c$ on the left side does not have a coefficient of 4, whereas the right side does. Therefore, the original statement $\log _3\left(c d^4\right)=4 \log _3 c+4 \log _3 d$ is False. It seems there might be a misunderstanding or a typo in the statement, as the power of $d$ within the logarithm should only affect the logarithm of $d$, not the logarithm of $c$. A correct application of the properties would yield $\log _3 c + 4 \log _3 d$. This exercise highlights the importance of precisely applying each rule.

Statement 2: $\frac{3}{4}\left(\ln a+\ln b\right)=\ln \sqrt[4]{a^3 b^3}$

Moving on to our second statement, we have $\frac{3}{4}\left(\ln a+\ln b\right)=\ln \sqrt[4]{a^3 b^3}$. This statement involves natural logarithms (ln), but the properties we will use apply to all bases of logarithms. The relevant properties here are again the product rule and the power rule. Let's start by simplifying the expression inside the parentheses on the left side: $\ln a+\ln b$. According to the product rule for logarithms, $\ln a+\ln b = \ln(ab)$.

Now, our left side becomes $\frac{3}{4}\ln(ab)$. We can use the power rule in reverse here. The power rule states that $n \log_b x = \log_b(x^n)$. Applying this, we can bring the coefficient $\frac{3}{4}$ as an exponent to the argument of the logarithm: $\frac{3}{4}\ln(ab) = \ln\left((ab)^{\frac{3}{4}}\right)$.

Let's simplify the argument $(ab)^{\frac{3}{4}}$. Using the exponent rule $(xy)^n = x^n y^n$, we get $(ab)^{\frac{3}{4}} = a^{\frac{3}{4}} b^{\frac{3}{4}}$. So, the left side is equal to $\ln\left(a^{\frac{3}{4}} b^{\frac{3}{4}}\right)$.

Now, let's look at the right side of the original statement: $\ln \sqrt[4]{a^3 b^3}$. We can rewrite the fourth root using fractional exponents. Recall that $\sqrt[n]{x} = x^{\frac{1}{n}}$. Therefore, $\sqrt[4]{a^3 b^3} = \left(a^3 b^3\right)^{\frac{1}{4}}$.

Using the exponent rule $(xy)^n = x^n y^n$, we get $\left(a^3 b^3\right)^{\frac{1}{4}} = \left(a^3\right)^{\frac{1}{4}} \left(b^3\right)^{\frac{1}{4}}$.

Using the exponent rule $(x^m)^n = x^{mn}$, we have $\left(a^3\right)^{\frac{1}{4}} = a^{3 \times \frac{1}{4}} = a^{\frac{3}{4}}$ and $\left(b^3\right)^{\frac{1}{4}} = b^{3 \times \frac{1}{4}} = b^{\frac{3}{4}}$.

So, the right side of the statement simplifies to $\ln\left(a^{\frac{3}{4}} b^{\frac{3}{4}}\right)$.

Comparing the simplified left side, $\ln\left(a^{\frac{3}{4}} b^{\frac{3}{4}}\right)$, with the simplified right side, $\ln\left(a^{\frac{3}{4}} b^{\frac{3}{4}}\right)$, we see that they are identical. Thus, the statement $\frac{3}{4}\left(\ln a+\ln b\right)=\ln \sqrt[4]{a^3 b^3}$ is True. This demonstrates how the power rule for logarithms and exponent rules can be used in conjunction to manipulate logarithmic expressions.

Statement 3: $3 \ln e-2 \ln e=1$

Finally, let's evaluate the third statement: $3 \ln e-2 \ln e=1$. This statement involves the natural logarithm of the base $e$ (Euler's number). A key property of logarithms is that $\log _b b = 1$. Since the natural logarithm is the logarithm with base $e$ (i.e., $\ln x = \log _e x$), we know that $\ln e = \log _e e = 1$. This is because $e$ raised to the power of 1 equals $e$ ($e^1 = e$).

Now we can substitute the value of $\ln e$ into the given equation. The left side of the equation is $3 \ln e-2 \ln e$. Substituting $\ln e = 1$, we get $3(1) - 2(1)$.

Performing the arithmetic: $3 - 2 = 1$.

So, the left side of the equation simplifies to 1. The right side of the equation is also 1. Since the left side equals the right side, the statement $3 \ln e-2 \ln e=1$ is True. This is a straightforward application of the fundamental property that the logarithm of the base is always 1. It's a good reminder that sometimes the simplest properties are the most powerful.

Conclusion: Mastering Logarithm Properties

Throughout this discussion, we've applied several core logarithm properties: the product rule ($\log _b(xy) = \log _b x + \log _b y$), the power rule ($\log _b(x^n) = n \log _b x$), and the fundamental property that $\log _b b = 1$. We also saw how these interact with exponent rules to manipulate expressions. The first statement was found to be False due to an incorrect application of the power rule. The second statement was True, showcasing a clever combination of the product and power rules. The third statement was True, relying on the basic definition of a logarithm involving the base $e$.

Grasping these logarithm properties is not just about passing a test; it's about building a strong foundation for more advanced mathematical concepts in calculus, differential equations, and beyond. Practice is key! Try working through various problems that require the use of these properties. The more you practice, the more intuitive they will become. Remember to always pay close attention to the details of each property and how it is applied.

For further exploration and practice on logarithm properties, you might find the resources at Khan Academy very helpful. They offer comprehensive explanations and practice exercises that can solidify your understanding. Another excellent resource for in-depth mathematical learning is Brilliant.org, which provides interactive courses on various mathematical topics, including logarithms.