91Ó°ÊÓ

Logarithms are the inverse operations to exponentiation, meaning they undo the action of raising a number to a power. In the context of functions, logarithms help us understand growth, decay, and distributions.
For a function like \(f(x) = \log(7-x)\), this means that we are seeking the power to which a base number (often 10, as in log base 10) needs to be raised to obtain the inner expression \(7-x\).
Understanding logarithms in a function involves recognizing that they permit only positive input values. That's because a logarithm of zero or a negative number is undefined in the realm of real numbers.
This understanding is vital because it defines where the function makes sense and where it doesn't.

When dealing with logarithmic functions, inequalities become crucial. They help determine the allowable inputs (domain). For the function \(f(x) = \log(7-x)\), we derive the inequality \(7-x > 0\).
This is because logarithms are only defined for positive arguments.
To solve such inequalities, we need to isolate \(x\) on one side. Subtracting \(7\) from both sides gives \(-x > -7\).
Inequalities involving a negative on the variable require us to flip the inequality sign when multiplying or dividing by negative numbers. Therefore, \(x < 7\).
Thus, all values less than 7 are satisfactory inputs for this function.

The domain of a function refers to the complete set of possible input values (\(x\) values) that the function can accept. In essence, it gives us the boundary of a function's valid inputs.
For a logarithmic function like \(f(x) = \log(7-x)\), the requirement is that \(7-x\) must be greater than zero, making \(x < 7\) the domain of this function.
This tells us that the input \(x\) can include any number less than 7. As a result, \(x\) cannot equal 7 or any number greater than 7, since such values would make the logarithm's argument zero or negative, both of which are invalid.
Understanding the function's domain is crucial, as it ensures we only work within the boundaries where the function is mathematically sound.