Problem 143 - College Algebra Essentials

Question

Without using a calculator, find the exact value of $$\frac{\log _{3} 81-\log _{\pi} 1}{\log _{2 \sqrt{2}} 8-\log 0.001}$$

Step-by-Step Solution

Answer

The exact value of the expression is \( \frac{2}{3} \)

Step 1: Simplify the Logarithms

Start by simplifying the logarithms. Using the definition of a logarithm, \(\log_{b} a=n\) means that \(b^n=a\). This rule can be applied to reduce the first term to \(4\) since \(\log_{3} 81 = 4\), because \(3^4 = 81\). The second term \(\log_{π} 1 = 0\) because \(π^0 = 1\). The third term can be simplified using the power rule \(\log_{b} (a^n) = n * log_{b} a\), so \(\log_{2\sqrt{2}}8 = \log_{2}(2^3)\), which reduces to \(3\). The last term \(\log (0.001)=-3\) because \(10^{-3} = 0.001\).

Step 2: Apply the Quotient Rule

The terms in the numerator and denominator are subtracted, which can be simplified by applying the quotient rule \(\log_{b} (a/c)= log_{b} a - log_{b} c\). The original problem is now simplified to \[ \frac{4-0}{3-(-3)} \]

Step 3: Calculate the Final Result

Finally, calculate the fraction \[ \frac{4}{6} = \frac{2}{3} \] Each step simplified the expression by applying a log rule. The final result is \(\frac{2}{3}\)