Without using a calculator, find the exact value of: [log₃ 81 - log_𝝅 1]/[log_2√2 8 - log 0.001]

Verified step by step guidance

1. Rewrite each logarithm in terms of simpler expressions or known values. For example, recognize that 81 can be written as a power of 3, and 8 can be written as a power of 2.
2. Use the logarithm power rule: $\log {\mathrm{a}}_b{c}^n=\; n\log {\mathrm{a}}_{b}c$, to simplify the logarithms of powers.
3. Apply the logarithm subtraction rule: $\log {\mathrm{a}}_{b}x-\log {\mathrm{a}}_{b}y=\log {\mathrm{a}}_b\frac{x}{y}$, to combine the logarithms in the numerator and denominator where possible.
4. Convert all logarithms to a common base if necessary, using the change of base formula: $\log {\mathrm{a}}_{b}c=\frac{\log }{d}\frac{\log }{d}$, where $d$ is a convenient base such as 10 or e.
5. After simplifying numerator and denominator separately, divide the two results to find the exact value of the original expression.

Key Concepts

Properties of Logarithms

Logarithms have key properties such as the product, quotient, and power rules that simplify expressions. For example, log_b(x) - log_b(y) = log_b(x/y), and log_b(x^k) = k * log_b(x). These properties allow rewriting and simplifying complex logarithmic expressions without a calculator.

Change of Base Formula

The change of base formula, log_a(b) = log_c(b) / log_c(a), allows converting logarithms with any base to a common base, often base 10 or e. This is useful for comparing or simplifying logarithms with different bases, especially when exact values are needed without a calculator.

Evaluating Logarithms of Powers and Roots

Understanding how to evaluate logarithms of numbers expressed as powers or roots is essential. For example, 81 = 3^4, √2 = 2^(1/2), and 8 = 2^3. Recognizing these forms helps rewrite logarithms in terms of their bases and exponents, enabling exact calculation of their values.