Verified Answer :
- Convert \(5^3 = 125\) to logarithmic form by mapping the base, exponent, and result to the form \(\log_a c = b\), resulting in \(\log_5 125 = 3\).
- Evaluate \(\log_{10} 0.001\) by setting it to \(x\) and converting to the exponential form \(10^x = 0.001\).
- Simplify \(0.001\) as \(\frac{1}{1000} = 10^{-3}\), which directly gives \(x = -3\).
- The final results are \(\log_5 125 = 3\) and \(\log_{10} 0.001 = -3\), summarized as \(\boxed{\log_5 125 = 3; -3}\).
Explanation
- Problem Analysis
The problem asks for two tasks: rewrite an exponential expression in logarithmic form and evaluate a logarithmic expression. We are given the definition of the relationship between exponential and logarithmic forms: if \(a^b = c\), then \(\log_a c = b\), for \(a > 0\), \(a eq 1\), and \(c > 0\). We will apply this to both parts of the problem.
- Converting to Logarithmic Form
To convert the expression \(5^3 = 125\) into logarithmic form, we identify the components based on the definition: the base \(a\) is \(5\), the exponent \(b\) is \(3\), and the result \(c\) is \(125\). Substituting these values into \(\log_a c = b\), we get \(\log_5 125 = 3\).
- Evaluating the Logarithm
To evaluate \(\log_{10} 0.001\), let \(x = \log_{10} 0.001\). Using the definition of logarithms, this is equivalent to solving \(10^x = 0.001\). Since \(0.001 = \frac{1}{1000} = \frac{1}{10^3}\), we can express this fraction using a negative exponent as \(10^{-3}\). Comparing \(10^x = 10^{-3}\), it follows that \(x = -3\). Therefore, \(\log_{10} 0.001 = -3\).
Examples
Logarithms are widely used in fields that deal with exponential growth or decay, such as measuring the intensity of sounds (decibels) or the magnitude of earthquakes (Richter scale). For example, if you know the power of an earthquake or the intensity of a sound compared to a baseline, you can use logarithms to determine the magnitude or the decibel level. Understanding how to switch between exponential and logarithmic forms helps solve equations where the variable is located in the exponent, which is a common occurrence in chemistry (pH levels), finance (compound interest), and biology (population growth models).