Use a calculator to find the common logarithm.
$\log (-83)$
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A. The answer is $\square$ .
(Round to four decimal places as needed.)
B. The answer does not exist.
Answer (1)
The problem asks for the common logarithm of -83, lo g ( − 83 ) .
Logarithms are only defined for positive real numbers.
Since -83 is negative, lo g ( − 83 ) does not exist.
The answer is that the logarithm does not exist.
Explanation
Understanding the Problem We are asked to find the common logarithm of -83, which is denoted as lo g ( − 83 ) . The common logarithm is the logarithm to base 10. We need to determine if the logarithm exists for this value.
Logarithm Definition Recall that the logarithm function (whether it's the common logarithm, the natural logarithm, or any other base logarithm) is only defined for positive real numbers. This is because the logarithm function asks the question: 'To what power must we raise the base to obtain the given number?' For example, lo g 10 ( 100 ) = 2 because 1 0 2 = 100 . However, there is no real number we can raise 10 to, to get a negative number like -83.
Conclusion Since -83 is a negative number, the common logarithm lo g ( − 83 ) does not exist within the realm of real numbers. Therefore, the correct choice is B.
Examples
Logarithms are used in many real-world applications, such as measuring the intensity of earthquakes (Richter scale) or the loudness of sound (decibels). These scales use logarithms because they allow us to represent a wide range of values in a more manageable way. However, these applications always deal with positive values, as the logarithm of a non-positive number is undefined in the real number system. Understanding the domain of logarithmic functions is crucial in these contexts.
