Negative Numbers and Logarithms: Is it really wrong?

[Hepic]

I know that in logarithms we can not set as base a negative number,but look at this(in the brackets I will put the base.): log(-2)-8=3 Mathematics say that is wrong,but why?
If we tell -2^3=-8 we have a correct result.
So? Thank you!

 

[arildno]

That is certainly true.

But:
You'll only be able to find such matches for some INTEGER values.

With any positive number distinct from 1 as chosen as base, we can, with the associated logarithm function represent EVERY positive number as a power of our base.

Having a negative number as your base, however, you cannot gain this type of general usage, that is, a continuous logarithm function covering, say, all the negative numbers is impossible to construct.
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Thus, for simplicity, we say that you can't have a negative number as your base.

 

[Hepic]

That was very clear answer. I hope to be many other guys like you in this forum

 

[arildno]

You are welcome! There are many such guys here.

 

[CellCoree]

I can understand the confusion around using negative numbers in logarithms. However, it is important to note that the rules and principles of mathematics are based on logical and consistent reasoning. In the case of logarithms, the base must always be a positive number because the concept of logarithms is based on the idea of repeated multiplication. When we take the logarithm of a number, we are essentially asking "what power do I need to raise the base to in order to get this number?" This concept only makes sense when the base is positive, as negative numbers do not have a defined root or power.

While it may seem like using -2^3= -8 follows the same logic as log(-2)-8=3, it is important to understand the fundamental differences between these operations. When we take the logarithm of a number, we are finding the exponent that would give us that number when raised to the base. In the case of -2^3, we are performing the operation of exponentiation, not taking the logarithm. Therefore, the rules and principles for logarithms do not apply in this situation.

In conclusion, it is not "wrong" to use negative numbers in mathematics, but it is important to understand the principles and rules that govern each operation. In the case of logarithms, the base must always be positive in order for the concept to make sense and be consistent with the principles of mathematics.