What is the approximate value of this logarithmic expression?
$\log_8 24$
The problem asks for the approximate value of the logarithmic expression \(\log_8 24\). To find this, we can use the properties of logarithms. First, notice that \(24\) can be written as \(8 \times 3\). Using the property of logarithms \(\log_b(xy) = \log_b x + \log_b y\), we can expand the expression:
\[\log_8 24 = \log_8 (8 \times 3) = \log_8 8 + \log_8 3.\]
Since \(\log_8 8 = 1\), the expression simplifies to:
\[1 + \log_8 3.\]
Next, we need to evaluate \(\log_8 3\). We can use the change of base formula, \(\log_b x = \frac{\log_k x}{\log_k b}\), to change the base from \(8\) to \(2\):
\[\log_8 3 = \frac{\log_2 3}{\log_2 8}.\]
Since \(8 = 2^3\), we know \(\log_2 8 = 3\). Thus,
\[\log_8 3 = \frac{\log_2 3}{3}.\]
Using a standard calculation for \(\log_2 3\), which is approximately \(1.585\), we find:
\[\log_8 3 \approx \frac{1.585}{3} \approx 0.5283.\]
Now, substitute the value of \(\log_8 3\) back into the simplified expression from the first step:
\[\log_8 24 = 1 + \log_8 3 \approx 1 + 0.5283 = 1.5283.\]
Comparing this result to the given options (\(1.38, 1.53, 0.48, 0.90\)), the best approximation is \(1.53\).
Logarithms are widely used in fields like acoustics and chemistry. For example, the loudness of a sound in decibels is measured using a logarithmic scale, and the pH of a chemical solution is the negative logarithm of the hydrogen ion concentration. Understanding how to approximate logarithmic values allows scientists and engineers to perform quick estimates when evaluating signal levels or chemical acidity in the field without needing a calculator.