Are the following statements true ?
log(-3) + log(-4) = log12
-log3 - log4 = -log12
Explain why or why not
The first equation is not true because there is no such thing as the log of a negative number. No positive number (base) raised to any power can be negative.
The second equation is equivalent to
log 3 + log4 = log12, which is true, since log12 = log(3x4)
To determine whether the given statements are true or not, let's break it down step by step:
Statement 1: log(-3) + log(-4) = log12
In mathematics, the logarithm function is defined only for positive real numbers. It is not defined for negative numbers or zero. Therefore, trying to calculate the logarithm of negative numbers is not valid.
In this case, log(-3) and log(-4) are not defined. So, the left side of the equation is undefined. On the other hand, the right side of the equation, log12, is defined and represents the logarithm of the positive number 12.
Since the left side is undefined and the right side is defined, Statement 1 is not true.
Statement 2: -log3 - log4 = -log12
This statement involves the negative sign applied to logarithms. When we have a negative sign in front of a logarithm, it indicates that the number being passed into the logarithm is reciprocal to the positive number.
For example, -log3 is equivalent to log(1/3), and -log4 is equivalent to log(1/4).
By applying this definition to the statement, we get:
-log3 - log4 = log(1/3) - log(1/4)
Using the properties of logarithms, we know that subtracting logarithms is equivalent to dividing the numbers inside the logarithms:
log(1/3) - log(1/4) = log((1/3) / (1/4))
Simplifying further,
log((1/3) / (1/4)) = log((4/3))
Now, we can compare -log12 and log((4/3)):
-log12 = log(1/12)
log((4/3)) ≠ log(1/12)
Therefore, -log3 - log4 ≠ -log12.
Hence, Statement 2 is also not true.
In summary, neither of the statements is true.