Question

Find the inverse of the function. \(y=\log _2 x\)

Short answer

The inverse of the function \(y=\log _2 x\) is \(y=2^x\).

Step-by-step solution

Understand inverse functions

The inverse of a function pairs every input of the original function with the corresponding output of the original function.

Interchange x and y in the equation

This is an integral part of finding the inverse of the function. Interchanging x and y helps to reverse the role of input and output. Thus, the equation becomes \(x=\log _2 y\).

Solve for the new y

Next step is to solve the equation for the new y. In this case, the inverse of a logarithmic function is an exponential function, so we can write the equation as \(2^x = y\). Therefore, the function inverse to \(y=\log _2 x\) is \(y=2^x\).

Key concepts

Logarithmic Functions

Logarithmic functions come into play whenever we deal with problems involving exponential growth or decay. Think of a logarithm as the opposite of an exponent. It tells us what exponent we need to use to get a particular number. For instance, with base 2, the logarithm gives us the power to which 2 must be raised to produce a given number. In the problem, the function is given by \(y = \log_2 x\). This means that for any number \(x\), \(\log_2 x\) gives you the power to which the base 2 needs to be raised to result in \(x\). Logarithms make it easier to solve equations that have exponential growth or decay by converting multiplication into addition, which is simpler to handle.

• Logarithms are the inverse of exponentials.
• They are essential in simplifying complex exponential equations.
• Understanding the base is crucial as it affects the result.

Exponential Functions

Exponential functions are a key concept in many applications, from natural phenomena like population growth to financial calculations involving compound interest. An exponential function can be expressed as \(y = b^x\), where \(b\) is the base and \(x\) is the exponent. In the solution provided to find the inverse of \(y = \log_2 x\), once the roles of \(x\) and \(y\) are reversed, we solve for the new \(y\) to get \(y=2^x\). This indicates that exponential functions and logarithmic functions are closely interconnected.

• Exponential functions involve raising a base number to a certain power.
• They are the inverse of logarithmic functions.
• Serve to model growth or decay scenarios effectively.

Function Inverse

The inverse of a function is essentially a reverse process. It reverses what the original function does. To find the inverse, you need to swap the inputs and outputs. In the context of the exercise, this means interchanging \(x\) and \(y\) in the equation \(y = \log_2 x\), leading to \(x = \log_2 y\). To solve for \(y\), we understand that the inverse of a logarithmic function is exponential, and hence, convert it to \(y = 2^x\). This gives us a clear function that undoes the logarithmic operation.

• Finding an inverse function involves swapping variables.
• The inverse reverses the original function's effect.
• Helps in solving equations that can be reversed.

Solving Equations

Solving equations is a fundamental aspect of mathematics. It involves finding values of variables that make an equation true. In solving equations involving logarithms and exponentials, understanding how to manipulate these functions is key. With the inverse function problem at hand, after interchanging \(x\) and \(y\), we then solve for \(y\). This requires recognizing that logarithmic equations can be transformed into exponential equations, allowing for easier solution. This method is particularly useful in handling more complex expressions and finding unknowns efficiently.

• Equation solving requires finding values that satisfy the equation.
• Transformations between logarithms and exponentials simplify solutions.
• Essential for finding inverse functions or reacting to changes in variables.