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

Ch. 4 - Exponential and Logarithmic Functions

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Blitzer 8th Edition

Ch. 4 - Exponential and Logarithmic Functions

Problema 143

Capitolo 5, Problema 143

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

Guida verificata passo dopo passo

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.

Use the logarithm power rule:

\log a_{b} c^{n} = n \log a_{b} c

, to simplify the logarithms of powers.

Apply the logarithm subtraction rule:

\log a_{b} x - \log a_{b} y = \log a_{b} \frac{x}{y}

, to combine the logarithms in the numerator and denominator where possible.

Convert all logarithms to a common base if necessary, using the change of base formula:

\log a_{b} c = \frac{\log}{d} \frac{\log}{d}

, where

d

is a convenient base such as 10 or e.

After simplifying numerator and denominator separately, divide the two results to find the exact value of the original expression.

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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.

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Change of Base Property

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.

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Change of Base Property

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.

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