how do you express the inverse of exponential equation
To express the inverse of an exponential equation, you will need to follow these steps:
1. Start with the given exponential equation and write it in the form y = f(x), where y represents the dependent variable and x represents the independent variable.
2. Swap the roles of x and y in the equation. This means replacing each occurrence of x with y and replacing each occurrence of y with x.
3. Solve the resulting equation for y. In other words, rearrange the equation to isolate y on one side.
4. The resulting equation is the inverse of the original exponential equation.
Let's take an example to better understand this process.
Example: If the original exponential equation is y = 2^x, we want to find its inverse.
Step 1: Start with the given equation: y = 2^x.
Step 2: Swap the roles of x and y: x = 2^y.
Step 3: Solve for y: To do this, we need to take the logarithm (base 2) of both sides. Applying the logarithm, we have log2(x) = log2(2^y), which simplifies to y = log2(x).
Step 4: The resulting equation, y = log2(x), is the inverse of the original exponential equation, y = 2^x.
Remember, the inverse of an exponential equation swaps the roles of the variables and involves taking the logarithm of both sides.