Questions: Simplify the expression log5(250). (Note log(10)/log(5) ≈ 1.4.)

Solution Steps

We are given the expression \(\log_{5}(250)\) and need to simplify it using the approximation \(\frac{\log(10)}{\log(5)} \approx 1.4\). We are also given multiple-choice options to select the correct answer.

The change of base formula for logarithms states that:

\[ \log_{b}(a) = \frac{\log_{c}(a)}{\log_{c}(b)} \]

Using this formula, we can express \(\log_{5}(250)\) in terms of common logarithms (base 10):

\[ \log_{5}(250) = \frac{\log(250)}{\log(5)} \]

We can estimate \(\log(250)\) using the properties of logarithms. Since \(250 = 2.5 \times 100\), we have:

\[ \log(250) = \log(2.5 \times 100) = \log(2.5) + \log(100) \]

We know that \(\log(100) = 2\) because \(100 = 10^2\). We approximate \(\log(2.5)\) using the fact that \(\log(2.5) \approx \log(2) + \log(1.25)\). For simplicity, let's assume \(\log(2.5) \approx 0.4\).

Thus:

\[ \log(250) \approx 0.4 + 2 = 2.4 \]

Now, substitute the estimated values into the change of base formula:

\[ \log_{5}(250) = \frac{\log(250)}{\log(5)} \approx \frac{2.4}{\log(5)} \]

Given \(\frac{\log(10)}{\log(5)} \approx 1.4\), we can solve for \(\log(5)\):

\[ \log(5) \approx \frac{\log(10)}{1.4} = \frac{1}{1.4} \approx 0.7143 \]

Substitute this back into the expression:

\[ \log_{5}(250) \approx \frac{2.4}{0.7143} \approx 3.36 \]

Final Answer