Express The Following Logarithms (to The Base 10) In Terms Of Logarithms Of Prime Integers.(a) Log ⁡ ( 50 ) 2 / 3 ( 30 ) 2 / 3 \log \frac{(50)^{2/3}}{(30)^{2/3}} Lo G ( 30 ) 2/3 ( 50 ) 2/3 ​ (b) $\log \left[\frac{2}{13} \sqrt{(15)^2 + (12) 2} {2/2}\right]

Introduction

Logarithms are a fundamental concept in mathematics, and they play a crucial role in various branches of mathematics, including algebra, geometry, and calculus. In this article, we will explore the process of expressing logarithms in terms of prime integers. We will start by understanding the properties of logarithms and then apply these properties to simplify the given expressions.

Properties of Logarithms

Before we dive into the main topic, let's recall some essential properties of logarithms:

• Product Rule: $\log(xy) = \log x + \log y$
• Quotient Rule: $\log\left(\frac{x}{y}\right) = \log x - \log y$
• Power Rule: $\log(x^y) = y\log x$
• Change of Base Formula: $\log_b a = \frac{\log_c a}{\log_c b}$

These properties will be instrumental in simplifying the given expressions.

Expressing Logarithms in Terms of Prime Integers

(a) $\log \frac{(50)^{2/3}}{(30)^{2/3}}$

To express this logarithm in terms of prime integers, we need to simplify the expression inside the logarithm.

First, let's factorize the numbers 50 and 30:

• $50 = 2 \cdot 5^2$
• $30 = 2 \cdot 3 \cdot 5$

Now, we can rewrite the expression as:

$\log \frac{(2 \cdot 5^2)^{2/3}}{(2 \cdot 3 \cdot 5)^{2/3}}$

Using the power rule, we can simplify the expression further:

$\log \frac{2^{2/3} \cdot 5^{4/3}}{2^{2/3} \cdot 3^{2/3} \cdot 5^{2/3}}$

Now, we can cancel out the common factors:

$\log \frac{5^{4/3}}{3^{2/3} \cdot 5^{2/3}}$

Using the quotient rule, we can simplify the expression further:

$\log 5^{4/3 - 2/3} - \log 3^{2/3}$

Simplifying the exponents, we get:

$\log 5^{2/3} - \log 3^{2/3}$

Using the power rule, we can rewrite the expression as:

$\frac{2}{3}\log 5 - \frac{2}{3}\log 3$

Now, we can factor out the common term:

$\frac{2}{3}(\log 5 - \log 3)$

Using the quotient rule, we can simplify the expression further:

$\frac{2}{3}\log \frac{5}{3}$

Therefore, the expression $\log \frac{(50)^{2/3}}{(30)^{2/3}}$ can be expressed in terms of prime integers as $\frac{2}{3}\log \frac{5}{3}$.

(b) $\log \left[\frac{2}{13} \sqrt{(15)^2 + (12)^2}^{2/2}\right]$

To express this logarithm in terms of prime integers, we need to simplify the expression inside the logarithm.

First, let's calculate the value inside the square root:

$(15)^2 + (12)^2 = 225 + 144 = 369$

Now, we can rewrite the expression as:

$\log \left[\frac{2}{13} \sqrt{369}^{2/2}\right]$

Using the power rule, we can simplify the expression further:

$\log \left[\frac{2}{13} \cdot 369^{1/2}\right]$

Now, we can rewrite the expression as:

$\log \left[\frac{2}{13} \cdot 3 \cdot 41^{1/2}\right]$

Using the product rule, we can simplify the expression further:

$\log 2 - \log 13 + \log 3 + \log 41^{1/2}$

Using the power rule, we can rewrite the expression as:

$\log 2 - \log 13 + \log 3 + \frac{1}{2}\log 41$

Now, we can factor out the common term:

$\log 2 + \frac{1}{2}\log 41 - \log 13 + \log 3$

Using the product rule, we can simplify the expression further:

$\log 2 + \frac{1}{2}\log 41 - \log 13 + \log 3$

Therefore, the expression $\log \left[\frac{2}{13} \sqrt{(15)^2 + (12)^2}^{2/2}\right]$ can be expressed in terms of prime integers as $\log 2 + \frac{1}{2}\log 41 - \log 13 + \log 3$.

Conclusion

In this article, we have explored the process of expressing logarithms in terms of prime integers. We have applied the properties of logarithms to simplify the given expressions and have expressed them in terms of prime integers. The process involves simplifying the expression inside the logarithm, using the properties of logarithms, and factoring out common terms. By following these steps, we can express logarithms in terms of prime integers, which is a fundamental concept in mathematics.

References

• [1] "Logarithms" by Math Is Fun
• [2] "Properties of Logarithms" by Purplemath
• [3] "Change of Base Formula" by Khan Academy

Further Reading

• "Introduction to Logarithms" by MIT OpenCourseWare
• "Logarithmic Functions" by Wolfram MathWorld
• "Properties of Logarithms" by Math Open Reference
  Logarithms in Terms of Prime Integers: Q&A =============================================

Introduction

In our previous article, we explored the process of expressing logarithms in terms of prime integers. We applied the properties of logarithms to simplify the given expressions and expressed them in terms of prime integers. In this article, we will answer some frequently asked questions related to logarithms in terms of prime integers.

Q: What is the difference between a logarithm and a prime integer?

A: A logarithm is the power to which a base number must be raised to produce a given value. A prime integer, on the other hand, is a positive integer that is divisible only by itself and 1.

Q: How do I simplify a logarithm in terms of prime integers?

A: To simplify a logarithm in terms of prime integers, you need to apply the properties of logarithms, such as the product rule, quotient rule, and power rule. You also need to factor out common terms and use the change of base formula if necessary.

Q: What is the change of base formula?

A: The change of base formula is a property of logarithms that allows you to change the base of a logarithm from one number to another. The formula is:

$\log_b a = \frac{\log_c a}{\log_c b}$

where $a$, $b$, and $c$ are positive real numbers.

Q: How do I apply the change of base formula?

A: To apply the change of base formula, you need to identify the base of the logarithm and the value inside the logarithm. You then need to plug these values into the formula and simplify.

Q: What are some common mistakes to avoid when simplifying logarithms in terms of prime integers?

A: Some common mistakes to avoid when simplifying logarithms in terms of prime integers include:

• Not applying the properties of logarithms correctly
• Not factoring out common terms
• Not using the change of base formula when necessary
• Not simplifying the expression inside the logarithm

Q: How do I check my work when simplifying logarithms in terms of prime integers?

A: To check your work when simplifying logarithms in terms of prime integers, you need to:

• Verify that you have applied the properties of logarithms correctly
• Check that you have factored out common terms correctly
• Verify that you have used the change of base formula correctly
• Simplify the expression inside the logarithm and verify that it is correct

Q: What are some real-world applications of logarithms in terms of prime integers?

A: Logarithms in terms of prime integers have many real-world applications, including:

• Cryptography: Logarithms are used to create secure encryption algorithms.
• Finance: Logarithms are used to calculate interest rates and investment returns.
• Science: Logarithms are used to calculate the pH of a solution and the concentration of a solution.
• Engineering: Logarithms are used to calculate the gain of an amplifier and the frequency response of a circuit.

Conclusion

In this article, we have answered some frequently asked questions related to logarithms in terms of prime integers. We have discussed the properties of logarithms, the change of base formula, and some common mistakes to avoid when simplifying logarithms in terms of prime integers. We have also discussed some real-world applications of logarithms in terms of prime integers.

References

• [1] "Logarithms" by Math Is Fun
• [2] "Properties of Logarithms" by Purplemath
• [3] "Change of Base Formula" by Khan Academy

Further Reading

• "Introduction to Logarithms" by MIT OpenCourseWare
• "Logarithmic Functions" by Wolfram MathWorld
• "Properties of Logarithms" by Math Open Reference