Question

without using a calculator, find the exact value of the following expression.\\(\frac{\log_{5}25 - \log_{\pi}1}{\log_{2\sqrt{2}}8 - \log 0.0001}\\)\\(\frac{\log_{5}25 - \log_{\pi}1}{\log_{2\sqrt{2}}8 - \log 0.0001} = \square\\) (simplify your answer.)

Explanation:

Step1: Simplify numerator terms

For \(\log_{5}25\), since \(5^2 = 25\), by the definition of logarithm \(\log_{a}a^b=b\), so \(\log_{5}25 = 2\).
For \(\log_{\pi}1\), since any non - zero number to the power of \(0\) is \(1\) (\(a^0 = 1,a
eq0\)), so \(\log_{\pi}1=0\).
Then the numerator \(\log_{5}25-\log_{\pi}1=2 - 0=2\).

Step2: Simplify denominator terms

First, simplify \(\log_{2\sqrt{2}}8\). Let \(y = \log_{2\sqrt{2}}8\), then \((2\sqrt{2})^y=8\). We know that \(2\sqrt{2}=2\times2^{\frac{1}{2}} = 2^{\frac{3}{2}}\) and \(8 = 2^3\). So \((2^{\frac{3}{2}})^y=2^3\), by the power - of - a - power rule \((a^m)^n=a^{mn}\), we have \(2^{\frac{3y}{2}}=2^3\). Then \(\frac{3y}{2}=3\), solving for \(y\) gives \(y = 2\).

Next, simplify \(\log_{10}0.0001\) (assuming it is a common logarithm with base \(10\)). Since \(0.0001=\frac{1}{10000}=10^{- 4}\), so \(\log_{10}0.0001=- 4\).

Then the denominator \(\log_{2\sqrt{2}}8-\log_{10}0.0001=2-(-4)=6\).

Step3: Calculate the fraction

Now we have the fraction \(\frac{\log_{5}25 - \log_{\pi}1}{\log_{2\sqrt{2}}8-\log_{10}0.0001}=\frac{2}{6}=\frac{1}{3}\).

Answer:

\(\frac{1}{3}\)