# Why can't logs have a negative base?

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
Mentor
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
Mentor
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.
Rumplestiltskin said:
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
Mentor
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
Staff Emeritus
Homework Helper
Gold Member
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 = x2

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.

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
Staff Emeritus
Homework Helper
Gold Member
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 .