Are the following statements true ?
log(-3) + log(-4) = log12
-log3 - log4 = -log12
Explain why or why not
I answered this already.
No, both of the statements are not true.
1) log(-3) + log(-4) = log12: This statement is not true because logarithms are undefined for negative numbers. The logarithm function is defined only for positive real numbers (greater than zero), so taking the logarithm of a negative number is not allowed. Therefore, log(-3) and log(-4) are undefined.
2) -log3 - log4 = -log12: This statement is also not true. Normally, when we subtract logarithms, it can be expressed as a division of the numbers inside the logarithms. However, in this case, the negative sign in front of log3 makes a difference. It is meant to indicate that we are taking the negative logarithm, also known as the antilogarithm. So, -log3 is the antilogarithm of 3, which is equal to 1/3. Therefore, the equation becomes:
-1/3 - log4 = -log12
Since -1/3 is not equal to -log12, the statement is false.
The logarithm of a negative number is undefined in the real number system. Logarithms can only be taken of positive numbers. Therefore, the first statement is not true. Log(-3) + log(-4) does not have a defined value.
Similarly, for the second statement, the negative sign in front of the logarithm cannot be distributed to the numbers inside the logarithm. So, -log3 - log4 cannot be simplified to -log12. Hence, the second statement is also not true.
To avoid using undefined logarithms, one can restrict the domain of the logarithm function to positive real numbers. In that case, the first and the second statements would be false.