Decoding Number Bases And Logarithms Solving $123_p = 38$ And Logarithmic Expressions

In this section, we embark on a journey to decipher the value of the base $p$ in the equation $123_p = 38$. This problem delves into the fascinating world of number systems, where numbers can be represented using different bases. The base of a number system determines the number of unique digits used to represent numbers. For instance, the decimal system we commonly use has a base of 10, employing digits from 0 to 9. However, other bases exist, such as binary (base 2), octal (base 8), and hexadecimal (base 16), each with its own set of digits.

To unravel the mystery of the base $p$, we need to understand how numbers are represented in different bases. In general, a number expressed in base $p$ can be written as a sum of powers of $p$, where each digit is multiplied by the corresponding power of $p$. For example, the number $123_p$ can be expanded as follows:

$$123_p = 1 \cdot p^2 + 2 \cdot p^1 + 3 \cdot p^0$$

Now, we are given that $123_p = 38$. Substituting the expanded form, we get:

$$1 \cdot p^2 + 2 \cdot p^1 + 3 \cdot p^0 = 38$$

Simplifying the equation, we have:

$$p^2 + 2p + 3 = 38$$

Rearranging the terms, we obtain a quadratic equation:

$$p^2 + 2p - 35 = 0$$

To solve this quadratic equation, we can employ various techniques, such as factoring, completing the square, or using the quadratic formula. In this case, factoring seems to be the most straightforward approach. We need to find two numbers that multiply to -35 and add up to 2. These numbers are 7 and -5. Therefore, we can factor the quadratic equation as follows:

$$(p + 7)(p - 5) = 0$$

This equation has two solutions:

$$p = -7 \text{ or } p = 5$$

Since the base of a number system cannot be negative, we discard the solution $p = -7$. Therefore, the value of the base $p$ is 5. In conclusion, by understanding the principles of number systems and solving the resulting quadratic equation, we have successfully determined the value of the base $p$ in the equation $123_p = 38$, which is 5. This exercise highlights the importance of number bases in representing numerical values and the mathematical techniques used to manipulate them.

In this segment, we delve into the realm of logarithms, exploring their properties and applying them to evaluate the expression $\operatorname{login} \sqrt{35} + \operatorname{login} \sqrt{2} - \operatorname{login} \sqrt{7}$. Logarithms are mathematical functions that express the power to which a fixed number, the base, must be raised to produce a given number. They are the inverse operation to exponentiation and play a crucial role in various fields, including mathematics, science, and engineering.

To evaluate the given logarithmic expression, we will leverage the fundamental properties of logarithms. These properties allow us to simplify complex logarithmic expressions and make them easier to handle. The key properties we will utilize are:

1. Product Rule: $\operatorname{login} (a \cdot b) = \operatorname{login} a + \operatorname{login} b$
2. Quotient Rule: $\operatorname{login} (a / b) = \operatorname{login} a - \operatorname{login} b$
3. Power Rule: $\operatorname{login} (a^n) = n \cdot \operatorname{login} a$

Applying these properties to the given expression, we can simplify it step by step. First, we use the power rule to rewrite the square roots as exponents:

$$\operatorname{login} \sqrt{35} + \operatorname{login} \sqrt{2} - \operatorname{login} \sqrt{7} = \operatorname{login} (35^{1/2}) + \operatorname{login} (2^{1/2}) - \operatorname{login} (7^{1/2})$$

Next, we apply the power rule again to bring the exponents outside the logarithms:

$$= (1/2) \cdot \operatorname{login} 35 + (1/2) \cdot \operatorname{login} 2 - (1/2) \cdot \operatorname{login} 7$$

Now, we can factor out the common factor of 1/2:

$$= (1/2) [\operatorname{login} 35 + \operatorname{login} 2 - \operatorname{login} 7]$$

Using the product rule, we can combine the first two logarithmic terms:

$$= (1/2) [\operatorname{login} (35 \cdot 2) - \operatorname{login} 7]$$

$$= (1/2) [\operatorname{login} 70 - \operatorname{login} 7]$$

Finally, we apply the quotient rule to combine the remaining logarithmic terms:

$$= (1/2) \cdot \operatorname{login} (70 / 7)$$

$$= (1/2) \cdot \operatorname{login} 10$$

Assuming "login" refers to the base-10 logarithm (log₁₀), we know that $\operatorname{login} 10 = \operatorname{log}_{10} 10 = 1$. Therefore,

$$= (1/2) \cdot 1$$

$$= 1/2$$

Thus, the value of the logarithmic expression $\operatorname{login} \sqrt{35} + \operatorname{login} \sqrt{2} - \operatorname{login} \sqrt{7}$ is 1/2. This problem showcases the power of logarithmic properties in simplifying complex expressions and arriving at a concise solution. By applying the product, quotient, and power rules, we efficiently reduced the expression to a simple numerical value. Logarithms are indispensable tools in mathematics and various scientific disciplines, enabling us to tackle intricate problems involving exponential relationships.

In this comprehensive exploration, we have successfully navigated through two intriguing mathematical problems. First, we decoded the base $p$ in the equation $123_p = 38$, revealing the value of $p$ to be 5. This journey underscored the significance of number systems and the methods used to convert between different bases. Then, we ventured into the realm of logarithms, skillfully evaluating the expression $\operatorname{login} \sqrt{35} + \operatorname{login} \sqrt{2} - \operatorname{login} \sqrt{7}$, ultimately arriving at the answer of 1/2. This exercise demonstrated the potency of logarithmic properties in simplifying complex expressions. These examples highlight the interconnectedness of mathematical concepts and the importance of mastering fundamental principles to solve a wide range of problems.