Why can't logs have a negative base?

[Rumplestiltskin]

I understand that taking logs of a negative number isn't possible because no number to any power produces a negative number. But why not a negative base?
Say, log_-10(100) = 2. Rewritten, -10² = 100, which is accurate.
You could suggest that you may as well just ignore the negative because -x² = x², but it's still weird that this shows up as a syntax error on a calculator.

 

[mfb]

Try to calculate log_-10(50)=x.
Or, alternatively, solve log_-10(x)=1.7. What is x=(-10)^1.7?

For integers (with the right sign) as logarithm values, it would work, but only for those.

 

[Mark44]

I understand that taking logs of a negative number isn't possible because no number to any power produces a negative number. But why not a negative base?
Say, log_-10(100) = 2. Rewritten, -10² = 100, which is accurate.

No, it isn't. $-10^2 = - 100$ because the exponent has a higher precedence than the negation sign. What you probably meant was $(-10)^2$, which is 100.

You could suggest that you may as well just ignore the negative because -x² = x², but it's still weird that this shows up as a syntax error on a calculator.

For the reason given above, $-x^2 \ne x^2$, unless x happens to be 0. If you want to square a negative number on a calculator, put parentheses around the number, with the exponent outside the parentheses.

 

[Mark44]

Moved from Homework sections, as this is more of a general question than a homework question. @Rumplestiltskin, be advised that if you post in the HW sections, youi need to use the homework template, not delete it as you apparently did.

 

[SammyS]

I understand that taking logs of a negative number isn't possible because no number to any power produces a negative number. But why not a negative base?
Say, log_-10(100) = 2. Rewritten, -10² = 100, which is accurate.
You could suggest that you may as well just ignore the negative because -x² = x², but it's still weird that this shows up as a syntax error on a calculator.

You should be more careful regarding order of operations.

-10² = -100

and

-x² = - (x²)

I assume you meant to write

(-10)² = 100

and

(-x)² = x²The main problem with having a negative base in a logarithm, is that there is a problem defining a real valued exponential function having a negative base.

 

[Rumplestiltskin]

Try to calculate log_-10(50)=x.
Or, alternatively, solve log_-10(x)=1.7. What is x=(-10)^1.7?

For integers (with the right sign) as logarithm values, it would work, but only for those.

Syntax error on calculator. When typed into google, (-10)^1.7 = 29.4590465 - 40.5468989 i. Woah. Still at a loss.

The main problem with having a negative base in a logarithm, is that there is a problem defining a real valued exponential function having a negative base.

Could you elaborate?

 

[SammyS]

Syntax error on calculator. When typed into google, (-10)^1.7 = 29.4590465 - 40.5468989 i. Woah. Still at a loss.Could you elaborate?

That's a complex number. Isn't that a problem for a real function?

Try (-10)^π .