Problem 35 - Precalculus with Limits

Question

Using Properties of Logarithms In Exercises \(21-36\) , find the exact value of the logarithmic expression without using a calculator. (If this is not possible, then state the reason.) $$\log _{5} 75-\log _{5} 3$$

Step-by-Step Solution

Answer

The exact value of the expression is 2.

Step 1: Recognize the property of functions

Here, we need to understand the Quotient Rule of Logarithms which states that \( \log_b(M/N)=\log_b(M)-\log_b(N) \) where b (base) is greater than 0 but not equal to 1.

Step 2: Apply the property

Applying this property, we have \( \log_5 (75/3) \), which simplifies to \( \log_5 (25) \).

Step 3: Calculate the logarithm

Finally, we calculate the value. By definition of logarithms, the base 5 raised to what power equals 25? The answer is 2, because \( 5^2 = 25 \). Therefore, \( \log_5(25) = 2 \).

Key Concepts

Quotient Rule of Logarithms

Understanding the quotient rule of logarithms is essential in simplifying expressions involving the division of two logarithmic terms with the same base. It states that the difference between two logarithms with the same base can be rewritten as the logarithm of the division of their respective arguments.

For instance, take the expression \(\log_b(M) - \log_b(N)\), where \(b\) is the base of the logarithm, and \(M\) and \(N\) are numbers greater than 0. According to the quotient rule, this expression can be simplified to \(\log_b\left(\frac{M}{N}\right)\).

• The base \(b\) must be positive and not equal to 1.
• Both \(M\) and \(N\) must be positive, as logarithms of non-positive numbers are not defined.

In the given exercise, applying this rule to \(\log_5(75) - \log_5(3)\) simplifies the expression to \(\log_5(25)\), as 75 divided by 3 equals 25. What simplification does is make calculations easier and often possible without a calculator, particularly when the numbers are perfect powers of the base, like in this case.

Logarithmic Expressions

Logarithmic expressions represent the power to which a base must be raised to yield a certain number. They are written in the form \(\log_b(x)\), where \(b\) is the base, and \(x\) is the number we're interested in. The expression answers the question: 'To what power must \(b\) be raised in order to get \(x\)?'

For example, if we have \(\log_5(125)\), it signifies the power that 5 must be raised to in order to equals 125. Since \(5^3 = 125\), \(\log_5(125) = 3\).

• \(\log_b(b) = 1\) because \(b^1 = b\).
• \(\log_b(1) = 0\) for any \(b\), because \(b^0 = 1\).
• The expression is undefined for \(x \leq 0\).

Logarithmic expressions are the inverse operations of exponents, making them incredibly useful for solving equations and problems involving exponential growth or decay.

Simplifying Logarithms

Simplifying logarithms involves reducing complex logarithmic expressions into a more manageable form, often using logarithmic properties. This allows easier computation and can help in solving logarithmic equations more efficiently.

To simplify a logarithm, like \(\log_b(M^k)\), you can use the Power Rule to pull the exponent \(k\) out front, resulting in \(k\cdot\log_b(M)\), assuming \(M\) is a positive real number. Simplification often involves applying a mix of the following logarithmic properties:

• The Product Rule: \(\log_b(MN) = \log_b(M) + \log_b(N)\)
• The Quotient Rule: \(\log_b\left(\frac{M}{N}\right) = \log_b(M) - \log_b(N)\)
• The Power Rule: \(\log_b(M^k) = k\cdot\log_b(M)\)

In the exercise given, \(\log_5(75) - \log_5(3)\) can be simplified through the Quotient Rule to \(\log_5(25)\), making it simpler to evaluate without a calculator. This is because 25 is a power of 5 (the base), meaning the logarithm becomes an easily identifiable integer, in this case, 2.