Problem 70 - Algebra 2

Question

Find the inverse of each function. $$ y=\log (x-2) $$

Step-by-Step Solution

Answer

The inverse of the function \(y = \log (x -2)\) is \(y = 10^{x} + 2\).

Step 1: Identify the Function

The provided function is \(y = \log (x -2)\). This is a logarithmic function.

Step 2: Interchange x and y

To find the inverse of a function, switch the places of y and x. This means that the function becomes \(x = \log (y -2)\).

Step 3: Solve for y

Rearrange the equation to solve for y. The inverse of the logarithmic function can be found by expounding both sides to base 10, resulting to the equation \(y = 10^{x} + 2\).

Key Concepts

Logarithmic Functions

Logarithmic functions are a unique type of mathematical function that help us understand exponential relationships. The basic idea is that a logarithm answers the question: "To what power must the base be raised, to produce a certain number?" A simple example is if we want to find out what power we need to raise 10 to, so that it equals 100. The answer is 2, because \(10^2 = 100\). This means \(\log_{10} 100 = 2\).
In the case of the function \(y = \log(x-2)\), the base is 10 because it is implied when no base is written. This function tells us how much the input \(x\), after adjusting by subtracting 2, must be scaled by the base to get \(y\). Logarithmic functions like this one help in solving for unknown exponents, especially in fields such as science and engineering.
Remember, when working with logarithmic functions, they are closely related to their exponential counterparts. Understanding this concept is key to both finding inverses and solving more complex problems.

Finding Inverse

Finding the inverse of a function allows us to reverse the roles of the inputs and outputs. It essentially answers the question: "If this transformation brings us to a particular output, what's the input that would bring us back?" To find an inverse, we generally swap the dependent and independent variables and then solve for the new dependent variable.
For example, starting with the equation \(y = \log(x-2)\), become \(x = \log(y-2)\) once we switch \(x\) and \(y\). Now, we need to solve for the new \(y\). Finding an inverse is like solving a puzzle that translates a complicated function back into its basic components, making complex ideas easier to understand.
It's important to check if the function is one-to-one to ensure the inverse exists. A one-to-one function is one where each input corresponds to a unique output. This is crucial otherwise, the inverse function might not be accurate.

Algebraic Manipulation

Algebraic manipulation involves rearranging equations to isolate a particular variable. This skills helps us in finding inverses and solving equations. It involves rules of mathematics such as balancing, expanding, and substituting terms.
In our case of finding the inverse of \(x = \log(y-2)\), we perform algebraic manipulation by exponentiating both sides to remove the logarithm. We raise 10 to the power of both sides, which gives us \(10^x = y - 2\). This step utilizes the fact that \( \log a = b \) means \( a = 10^b \) for a base 10 logarithm.
Once transformed, solving for \(y\) results in \(y = 10^x + 2\). This solution re-arrangement showcases clean algebraic manipulation and highlights how inverses of logarithmic functions often resolve into exponential forms. Understanding these steps is crucial for handling more complex functions in the future.