Problem 75 - Algebra and Trigonometry

Question

\(75-76\) A function \(f(x)\) is given. (a) Find the domain of the function \(f\) (b) Find the inverse function of \(f .\) $$ f(x)=\log _{2}\left(\log _{10} x\right) $$

Step-by-Step Solution

Answer

Domain: \( x > 1 \); Inverse: \( f^{-1}(x) = 10^{2^x} \).

Step 1: Understand the Function

The given function is \( f(x) = \log_{2}(\log_{10}x) \). This is a composition of two logarithmic functions. First, determine what values of \( x \) allow the expression inside each logarithm to be real.

Step 2: Determine the Inner Domain

The inner function is \( \log_{10}x \). The logarithm of a number is defined only for positive real numbers. Therefore, \( x > 0 \) must be true to ensure \( \log_{10}x \) is defined.

Step 3: Determine the Outer Domain

Next, consider the outer function \( \log_{2}(y) \) where \( y = \log_{10}x \). For \( \log_{2}(y) \) to be defined, \( y \) must be positive. Thus, \( \log_{10}x > 0 \).

Step 4: Solve for the Domain Condition

The inequality \( \log_{10}x > 0 \) implies that \( x > 10^{0} \) because logarithms base 10 of 1 equals 0. Thus, \( x > 1 \). Therefore, the domain of \( f(x) \) is \( x > 1 \).

Step 5: Consider Finding the Inverse

For the inverse, set \( y = f(x) = \log_{2}(\log_{10}x) \). Switch \( x \) and \( y \) to begin finding the inverse: \( x = \log_{2}(\log_{10}y) \).

Step 6: Isolate the Inner Logarithm Function

To isolate \( \log_{10}y \), rewrite the equation in exponential form: \( \log_{10}y = 2^x \).

Step 7: Solve for the Inverse Function

Now solve for \( y \) by rewriting \( \log_{10}y = 2^x \) in exponential form: \( y = 10^{2^x} \). Thus, the inverse function is \( f^{-1}(x) = 10^{2^x} \).

Step 8: Summary of Solutions

The domain of \( f(x) \) is \( x > 1 \), and the inverse function is \( f^{-1}(x) = 10^{2^x} \).

Key Concepts

Domain of a Function

The domain of a function is the set of all possible input values (usually denoted as \(x\)) that allow the function to operate without any mathematical errors. For logarithmic functions, the domain is determined by ensuring that the argument of the logarithm is a positive number. Logarithms are undefined for zero or negative numbers, so it is crucial to identify the values of \(x\) that make the expression inside the logarithm positive.
In the given function \( f(x) = \log_{2}(\log_{10}x) \), determining the inner domain involves the expression inside the first logarithm, which is \( \log_{10}x \). This requires \(x > 0\) since \( \log_{10}x \) must be a real number. Meanwhile, for the outer domain of \( \log_{2}(\log_{10}x) \) to be defined, \( \log_{10}x \) must be greater than 0.
This leads to solving \( \log_{10}x > 0 \), which means that \( x > 10^0 \) or simply \( x > 1 \). Therefore, the domain of the function \( f(x) \) is all real numbers greater than 1.

Logarithmic Functions

Logarithmic functions are essential in mathematics for their ability to inverse exponential operations. The general form of a logarithm is \( \log_{b}(x) \), representing the power to which the base \(b\) must be raised to produce the number \(x\).
In the case of \( f(x) = \log_{2}(\log_{10}x) \), we see a composition of logarithms. The inner function \( \log_{10}(x) \) converts \(x\) into a power of 10. Logarithms can only process positive numbers because you cannot raise 10 to any power to get a negative number or zero.
The outer function \( \log_{2}(y) \) processes this result \((y)\), transforming it into a power of 2. Each transformation depends on the output of the previous logarithmic function. This dependency implies the importance of ensuring each argument is strictly positive, hence defining the domain of \(x\) for the entire function.

Exponential Form

The exponential form is a way of expressing numbers in terms of a base raised to a power. It is central to solving expressions involving logarithmic functions. To convert a logarithmic equation into an exponential one, you switch the operation from logarithm to exponent.
For the inverse function in this exercise, we start with \(x = \log_{2}(\log_{10}y)\). To isolate \( \log_{10}y \), we express it in exponential form as \( \log_{10}y = 2^x \), meaning \( \log_{10}y \) equals 2 raised to the power of \(x\).
Once we have \( \log_{10}y = 2^x \), we can further solve for \(y\) by converting the equation into another exponential form: \( y = 10^{2^x} \). This is how the inverse function is derived: it transforms the input \(x\) by using nested exponents. It demonstrates the relationship between logarithms and exponents as natural inverses of each other.