For a certain base, $b, \log _b 8=3$ and $\log _b 0.5=-1$.
The value of $\log _b 4 b$ is [blank].
The problem provides two logarithmic equations involving a base \(b\): \(\log_b 8 = 3\) and \(\log_b 0.5 = -1\). We are asked to evaluate the expression \(\log_b(4b)\). First, we use the definition of a logarithm: \(\log_b x = y\) is equivalent to \(b^y = x\). Applying this to the first equation \(\log_b 8 = 3\), we have \(b^3 = 8\). Since \(8 = 2^3\), it follows that \(b^3 = 2^3\), which implies \(b = 2\). We can verify this with the second equation: \(\log_b 0.5 = -1\) means \(b^{-1} = 0.5\). Substituting \(b = 2\), we get \(2^{-1} = \frac{1}{2} = 0.5\), which is consistent. Thus, the base is \(b = 2\).
Now that we have established \(b = 2\), we need to compute \(\log_b(4b)\). Substituting \(b = 2\) into the expression, we get \(\log_2(4 \times 2) = \log_2 8\). Using the definition of the logarithm, we know \(\log_2 8 = 3\) because \(2^3 = 8\). Alternatively, we can use the property \(\log_b(xy) = \log_b x + \log_b y\). Thus, \(\log_2(4 \times 2) = \log_2 4 + \log_2 2\). Since \(4 = 2^2\), \(\log_2 4 = 2\). Since \(\log_2 2 = 1\), the result is \(2 + 1 = 3\).
Logarithms are essential in measuring scales that cover a wide range of values, such as the Richter scale for earthquake magnitude or the pH scale for acidity. For instance, if you know the base of a specific growth model, you can use logarithmic properties to predict how much time it will take for a quantity, like a population or an investment, to reach a certain target. This type of calculation helps scientists and financial analysts understand growth trends effectively.