Find the logarithm. [tex]$\log 10^{-6}$[/tex]
Answer (1)
The problem asks to find the logarithm of 1 0 − 6 .
Assume the base of the logarithm is 10, so we need to calculate lo g 10 1 0 − 6 .
Using the property lo g b b x = x , we find that lo g 10 1 0 − 6 = − 6 .
Therefore, the answer is − 6 .
Explanation
Understanding the Problem We are asked to find the logarithm of 1 0 − 6 , which is written as lo g 1 0 − 6 . Since the base of the logarithm is not specified, we assume it is the common logarithm, which has a base of 10. Therefore, we want to find the value of lo g 10 1 0 − 6 .
Applying Logarithm Properties To solve this, we use the property of logarithms that lo g b b x = x . In other words, the logarithm of a number to a certain base, where the number is the base raised to some power, is simply that power.
Calculating the Logarithm In our case, the base is 10, and the number is 1 0 − 6 . So, we have lo g 10 1 0 − 6 = − 6 .
Final Answer Therefore, the logarithm of 1 0 − 6 is -6.
Examples
Logarithms are used in many scientific fields, such as physics and chemistry, to simplify calculations and represent very large or very small numbers. For example, the pH scale, which measures the acidity or alkalinity of a solution, is based on logarithms. Similarly, the Richter scale, used to measure the magnitude of earthquakes, is also logarithmic. Understanding logarithms helps in interpreting these scales and making sense of the data they represent. Logarithmic scales allow us to represent a wide range of values in a more manageable way, making it easier to compare and analyze data.
