Can you not have negative logs?

[12345678]

Solve the equation log (x² - 10 ) – log (x) = 2log 3
Note that all the logs are to base 10, I just didn’t knowhow to write that down on the pc.
So using log laws I rewrote the equation
log ((x² - 10) / (x)) = log3²
So you can ignore the logs, giving you..
(x² - 10)/(x) = 9
x² - 10 = 9x
x² - 9x – 10 =0
Factorising this gives
(X – 10) * (X + 1) = 0
So X = 10 and X =-1
My question Is why isthe answer only x = 10? I did theworking above but only got 4/6 marks because the mark scheme credited onlygetting 10 as the answer.
Cheers.

 

[srmichael]

Solve the equation log (x² - 10 ) – log (x) = 2log 3
Note that all the logs are to base 10, I just didn’t knowhow to write that down on the pc.
So using log laws I rewrote the equation
log ((x² - 10) / (x)) = log3²
So you can ignore the logs, giving you..
(x² - 10)/(x) = 9
x² - 10 = 9x
x² - 9x – 10 =0
Factorising this gives
(X – 10) * (X + 1) = 0
So X = 10 and X =-1
My question Is why isthe answer only x = 10? I did theworking above but only got 4/6 marks because the mark scheme credited onlygetting 10 as the answer.
Cheers.

What is the domain for log(x)? ;-)

 

[Deleted member 4993]

By definition:

log_b(a) = c → b^c = a

If b & c are real positive numbers then b^c ≥ 0 → a ≥ 0(some people may argue b^c > 0 → a >0)

 

[12345678]

What is the domain for log(x)? ;-)

The question includes all information given and the base of the logs is 10.

 

[srmichael]

The question includes all information given and the base of the logs is 10.

I think you misunderstood my post. I wasn't asking you what the domain was for the problem, I was asking you, in general, what is the domain of a log function. To not string this thread out any longer, the domain of log(x) is x > 0. So that is why x cannot be -1.

 

[Quaid]

Can you not have negative logs?

The meaning of the phrase "negative log" is not clear.

If we take the base-10 logarithm of a Real number between zero and one, we'll get a negative value.

log(1/10) = -1

Is that a "negative log"?


But, you're trying to ask whether we can take the logarithm of a negative number, yes?

The answer is "sometimes"!


We cannot take the logarithm of a negative number, if we're restricted to the Real number system.

However, with complex-valued logarithms, we have log(-1) = i*π


So, maybe you can get away with telling your instructor that you had assumed complex-valued logarithms, when you answered the exercise. :wink:

Otherwise, memorize the general behavior of y = log(x), in the xy-plane, so that you can easily remember where log(x) is negative, where it's zero, where it's postive, and where it's not defined.

Cheers