Without using a calculator, find the exact value of: [log3 81 - log𝝅 1]/[log2√2 8 - log 0.001]

Verified step by step guidance

1. Step 1: Recognize that \( \log_3 81 \) can be simplified using the fact that 81 is a power of 3. Specifically, \( 81 = 3^4 \), so \( \log_3 81 = 4 \).
2. Step 2: Understand that \( \log_\pi 1 \) is 0 because any logarithm of 1 is 0, regardless of the base.
3. Step 3: Simplify the expression in the numerator: \( \log_3 81 - \log_\pi 1 = 4 - 0 = 4 \).
4. Step 4: For the denominator, recognize that \( \log_{\sqrt{2}} 8 \) can be rewritten using the change of base formula: \( \log_{\sqrt{2}} 8 = \frac{\log_2 8}{\log_2 \sqrt{2}} \). Since \( 8 = 2^3 \), \( \log_2 8 = 3 \). Also, \( \sqrt{2} = 2^{1/2} \), so \( \log_2 \sqrt{2} = \frac{1}{2} \). Therefore, \( \log_{\sqrt{2}} 8 = \frac{3}{1/2} = 6 \).
5. Step 5: Recognize that \( \log 0.001 \) is \( \log 10^{-3} = -3 \). Therefore, the denominator becomes \( 6 - (-3) = 6 + 3 = 9 \). Finally, the expression simplifies to \( \frac{4}{9} \).

Key Concepts

Logarithmic Properties

Logarithmic properties are rules that simplify the manipulation of logarithms. Key properties include the product rule (log_b(mn) = log_b(m) + log_b(n)), the quotient rule (log_b(m/n) = log_b(m) - log_b(n)), and the power rule (log_b(m^k) = k * log_b(m)). Understanding these properties is essential for simplifying expressions involving logarithms.

Change of Base Formula

The change of base formula allows the conversion of logarithms from one base to another, expressed as log_b(a) = log_k(a) / log_k(b) for any positive k. This is particularly useful when dealing with logarithms of bases that are not easily computable, enabling the evaluation of logarithmic expressions in a more manageable form.

Exponential and Logarithmic Relationships

Exponential and logarithmic functions are inverses of each other. For example, if b^y = x, then log_b(x) = y. This relationship is crucial for solving logarithmic equations and understanding how to manipulate logarithmic expressions, as it provides a foundation for converting between exponential and logarithmic forms.