Correct Logarithmic Conversions: A Step-by-Step Guide

Hey there, math enthusiasts! Today, we're diving deep into the fascinating world of logarithms and exploring the correctness of some logarithmic conversions. Logarithms can seem a bit intimidating at first, but with a clear understanding of their properties, you'll be able to tackle these problems with confidence. So, let's put on our thinking caps and figure out which of the following logarithmic conversions hold true. This article will break down each conversion, explaining the underlying principles and providing clear, step-by-step explanations. By the end, you’ll not only know the correct answers but also understand why they’re correct. Understanding these conversions is crucial for anyone delving into higher mathematics, physics, engineering, or any field that uses logarithmic scales.

Unpacking the Logarithmic Conversions

We have four logarithmic conversions to examine:

1. $\log _4(20)=\frac{\ln (20)}{\ln (4)}$
2. $\log _x(2)=\frac{\ln (2)}{\ln (x)}$
3. $\log _2(x)=\frac{\ln (2)}{\ln (x)}$
4. $\ln (x)=\frac{\log (x)}{\log (e)}$

Let's dissect each one individually, applying the fundamental principles of logarithmic conversions. We'll start with the change of base formula, a cornerstone in understanding these transformations.

The Change of Base Formula: Your Logarithmic Superpower

The change of base formula is the key to unlocking these logarithmic conversions. It states that for any positive numbers a, b, and x (where a ≠ 1 and b ≠ 1), the following holds true:

$\log_b(x) = \frac{\log_a(x)}{\log_a(b)}$

This formula allows us to convert a logarithm from one base (b) to another (a). It's like having a universal translator for logarithms! The most common bases we'll encounter are base 10 (common logarithm, denoted as log) and base e (natural logarithm, denoted as ln). The natural logarithm, often written as ln(x), is simply the logarithm to the base e, where e is Euler's number (approximately 2.71828). It pops up frequently in calculus and other areas of mathematics. Mastering this formula is essential for simplifying and solving logarithmic equations.

Conversion 1: $\log _4(20)=\frac{\ln (20)}{\ln (4)}$

Let's apply the change of base formula to the first conversion. We have $\log _4(20)$ on the left side. If we want to change the base to e (natural logarithm), we can use the formula:

$\log_4(20) = \frac{\log_e(20)}{\log_e(4)}$

Since $\log_e$ is the same as $\ln$, we can rewrite this as:

$\log_4(20) = \frac{\ln(20)}{\ln(4)}$

This exactly matches the right side of the given equation. Therefore, the first conversion is correct. This demonstrates a direct application of the change of base formula. By changing the base from 4 to e, we've successfully expressed the logarithm in terms of natural logarithms. This technique is invaluable when using calculators, as most calculators have built-in functions for natural logarithms.

Conversion 2: $\log _x(2)=\frac{\ln (2)}{\ln (x)}$

Now, let's tackle the second conversion: $\log _x(2)=\frac{\ln (2)}{\ln (x)}$. Again, we can use the change of base formula to convert the logarithm on the left side from base x to base e:

$\log_x(2) = \frac{\log_e(2)}{\log_e(x)}$

Replacing $\log_e$ with $\ln$, we get:

$\log_x(2) = \frac{\ln(2)}{\ln(x)}$

This matches the right side of the equation, so the second conversion is also correct. This conversion highlights the flexibility of the change of base formula. We were able to change the base from a variable x to the natural base e, demonstrating the formula's versatility in handling different bases.

Conversion 3: $\log _2(x)=\frac{\ln (2)}{\ln (x)}$

Let's examine the third conversion: $\log _2(x)=\frac{\ln (2)}{\ln (x)}$. Applying the change of base formula to the left side, we get:

$\log_2(x) = \frac{\ln(x)}{\ln(2)}$

Notice that this is the reciprocal of the right side of the given equation. The correct conversion should have $\ln(x)$ in the numerator and $\ln(2)$ in the denominator. Therefore, the third conversion is incorrect. This example underscores the importance of careful application of the change of base formula. A simple reversal of the numerator and denominator leads to an incorrect conversion.

Conversion 4: $\ln (x)=\frac{\log (x)}{\log (e)}$

Finally, let's analyze the fourth conversion: $\ln (x)=\frac{\log (x)}{\log (e)}$. Remember that $\ln(x)$ is the logarithm with base e, and $\log(x)$ is the common logarithm with base 10. We can rewrite the right side using the change of base formula, converting the base of the logarithm in the numerator from 10 to e:

$\frac{\log (x)}{\log (e)} = \frac{\log_{10}(x)}{\log_{10}(e)} = \log_e(x)$

Since $\log_e(x)$ is the same as $\ln(x)$, the equation holds true. So, the fourth conversion is correct. This conversion reinforces the connection between natural logarithms and common logarithms through the change of base formula. It demonstrates how a logarithm in base 10 can be expressed in terms of a logarithm in base e, and vice versa.

Conclusion: Mastering Logarithmic Conversions

In summary, after carefully analyzing each conversion using the change of base formula, we've determined that:

• Conversion 1 is correct. $\log _4(20)=\frac{\ln (20)}{\ln (4)}$
• Conversion 2 is correct. $\log _x(2)=\frac{\ln (2)}{\ln (x)}$
• Conversion 3 is incorrect. $\log _2(x) \neq \frac{\ln (2)}{\ln (x)}$
• Conversion 4 is correct. $\ln (x)=\frac{\log (x)}{\log (e)}$

Understanding and applying the change of base formula is crucial for working with logarithms. It allows you to convert logarithms between different bases, simplifying calculations and solving equations. By mastering this formula, you'll be well-equipped to tackle a wide range of logarithmic problems. Remember to practice these conversions and apply them in various contexts to solidify your understanding. Logarithms are a fundamental tool in mathematics, science, and engineering, and a solid grasp of their properties will undoubtedly benefit you in your academic and professional pursuits. Keep practicing, keep exploring, and you'll become a logarithmic whiz in no time!

For more information on logarithmic functions and their properties, you can visit Khan Academy's Logarithm section.