Question

What is the logarithm of a negative number?

Answer 1

Hint: We know that logarithm functions are the inverse of exponential functions. The inverse of the exponential function $x=a^y$ is$y=a^x$. We know the natural logarithm functions are defined only for $x>0$ . So the natural logarithm of a negative number is undefined. The complex logarithm functions are defined for negative numbers too.

Complete step by step solution:
Now the complex number is a number that can be expressed in the form of $a+\iota b$, where a and b are real numbers and $\iota$ is a symbol called the imaginary unit.
Logarithms of negative numbers are not defined in the real numbers in the same way that the square roots of negative numbers are not defined in real numbers. The logarithm of negative numbers is undefined. The logarithms of negative numbers are defined in complex numbers.
To understand the logarithm number in complex numbers let's take an example,${\log }_b(-x)$. Now to find the value of above we will first use the change of base rule and then we will convert this above example in natural logarithms to make it in a simple form.
$\Rightarrow {\log }_b(-x)={\displaystyle \frac{\ln (-x)}{\ln b}}$
Here $\ln (-x)$ is the same thing as $\ln (-1\cdot x)$ . We can exploit the addition property of logarithms, and separate this part into two separate logs:
$\Rightarrow {\log }_b(-x)={\displaystyle \frac{\ln x+\ln (-1)}{\ln b}}$
Now it is hard to get the value of $\ln (-1)$. But there is a very famous equation known as Euler's identity. The Euler`s identity is:
$\Rightarrow e^{i\pi }=-1$
Now, take ln on both side of the Euler`s identity, then we get
$\Rightarrow \ln e^{i\pi }=\ln (-1)$
Now by more simplifying the above equation by writing $\ln e^{i\pi }=\iota \pi$ by using the logarithm rule which is $\ln a^b=b\ln a$ , then we get
$\Rightarrow \iota \pi =\ln (-1)$
Now we will put this value in out equation which is ${\log }_b(-x)={\displaystyle \frac{\ln x+\ln (-1)}{\ln b}}$, then we get
$\Rightarrow {\log }_b(-x)={\displaystyle \frac{\ln x+\iota \pi }{\ln b}}$
Hence, we get the formula to find the logarithms of negative numbers.

Note: Logarithmic functions are used in many ways. It lets you work backward through a calculation. It also lets you undo exponential effects. Logarithms are a convenient way to express large numbers. The graph of the logarithm functions are always toward the positive. That's it is hard to find the logarithm of negative numbers. But by using complex numbers we are easily able to find the logarithm of negative numbers.