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Chapter 1: Problem 9

For \(b>0\) with \(b \neq 1,\) what are the domain and range of \(f(x)=\log _{b} x\) and why?

Short Answer

Expert verified
Question: Determine the domain and range of the logarithmic function \(f(x) = \log_b{x}\), where \(b > 0\) and \(b \neq 1\). Answer: The domain of the function is (0, +鈭�), and the range of the function is (-鈭�, +鈭�).

Step by step solution

01

Determine the domain of the function

Recall that the logarithmic function is the inverse operation of the exponential function. A logarithm given as \(\log_b{x}\) seeks the power or exponent (y-value) that must be applied to the base (b) to obtain the x-value. Therefore, in logarithmic functions, x-values must always be positive, as raising the base to any exponent will never result in a negative value. Thus, the domain of the function will consist of all positive real numbers. Mathematically, we can represent this as: Domain of \(f(x) = \log_b{x} = (0, +\infty)\)
02

Determine the range of the function

Since we know that the logarithm function is the inverse operation of the exponential function, it covers all the exponents that the base can be raised to. Since we are considering a general logarithm function with \(b > 0\) and \(b \neq 1\), the base can be raised to any exponent to generate a possible output value. This means that the range of the function will span all real numbers, both positive and negative. Mathematically, we can represent this as: Range of \(f(x) = \log_b{x} = (-\infty, +\infty)\)
03

Putting the answers together

So, for the logarithmic function \(f(x) = \log_b{x}\) where \(b > 0\) and \(b \neq 1\), we have: - Domain: \((0, +\infty)\), as the input values must always be positive - Range: \((-\infty, +\infty)\), as the logarithm function spans all possible exponents that the base can be raised to.

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