Why can't the base of a log be negative, How to find logarithm of a number with negative base
Why can’t the base of a log be negative? Or Can the base of a log be a negative number? Or
Can the negative base of a log function be defined?
All of these questions have the same meaning. So, the best way to get the answer to these questions is to put a negative base in log and see what happens?
Suppose y equals log-base-negative 2 of 3; which can be written as the natural log of 3 upon a natural log of 2 times negative one.
Further, the expression on the right-hand side can be written as the natural log of 3 upon the sum of the natural log of 2 and the natural log of negative one.
Now, we can write a negative one equals the sum of the cosine of pi and iota times sine of pi.
I am ignoring general form, here.
By using Euler's formula, we can write a negative one equals “e” to the “i times PI".
Taking the natural log on both sides of this equation, we get a natural log of negative one equals “I" times PI.
Which is a purely imaginary number.
If we substitute the value of the natural log of negative one, we get y equals natural log of 3 upon the sum of the natural log of 2 and iota times PI.
This implies; y is an imaginary number, which is often called a complex number.
When we talk about complex functions, it is fine. But in case of real-valued-functions; negative input is not defined at all.
So, what next? If you want to know “Why is log not defined for negative values”, then watch this.
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