Problem 65 - Algebra 2

Question

Find the inverse of each function. $$ y=\log _{0.5} x $$

Step-by-Step Solution

Answer

The inverse of the function \(y = \log_{0.5}(x)\) is \(y = 0.5^x\).

Step 1: Understand the expression

The literal interpretation of the given equation \(y=\log_{0.5}(x)\) is the following: what exponent should we raise 0.5 to, in order to get x. The inverse of the function will be the opposite process.

Step 2: Convert Logarithmic to Exponential Form

The process involves converting the logarithmic equation into exponential form. To do this, we use the fact that \(b^y = x\) is the inverse of \(\log_{b}(x) = y\). So, apply this to our equation, we get \(0.5^y = x\).

Step 3: Isolate x to form the inverse

A function and its inverse switch the roles of the \(x\) and \(y\) variables. So, we replace \(x\) with \(y\) and \(y\) with \(x\) in our new equation to isolate \(y\). This gives us \(0.5^x = y\).

Step 4: Formulate Final Answer

Our final answer is therefore, the inverse of the function \(y = \log_{0.5}(x)\) is \(y = 0.5^x\).

Key Concepts

Logarithmic Functions

Logarithmic functions help us understand situations where values grow or diminish exponentially. Let’s consider the function you might see:

• The function \(y = \log_{b}(x)\) tells us to what power we need to raise the base \(b\) to obtain \(x\).

Logarithms turn the process of multiplying into simpler addition and have wide applications in solving different kinds of equations including exponential equations.
They reverse the process of exponentiation, which gives them a natural role in defining inverse functions such as the one given in the exercise.
Whenever you see a problem with a log function like \(y = \log_{0.5}(x)\), remember that it tells us the exponent required on the base, \(0.5\), that gives \(x\). This concept will become clearer the more you practice, so don’t hesitate to try more examples on your own.

Exponential Functions

Understanding exponential functions is crucial when learning about logarithms, as they are closely linked. An exponential function is of the form \(y = b^x\), where you take a base \(b\) and raise it to a power \(x\).

• They model growth or decay processes such as population growth, radioactive decay, and interest calculations.
• For our particular problem, the base \(0.5\) indicates a decay rather than growth, as values will decrease as \(x\) increases.

When you invert a logarithmic function, you arrive at an exponential function.
The basic role reversal of variables inverting the functions aligns with this principle.
Thus, what starts as a log problem often requires understanding exponential forms like \(y = b^x\) for complete solutions.

Function Inversion

Function inversion revolves heavily around switching roles between inputs and outputs.
In mathematics, the inverse of a function \(f(x)\) is another function that reverses the effect of \(f(x)\).

• For instance, if you start from \(x\), apply \(f\), then apply the inverse \(f^{-1}\), you end looped back, arriving at your starting point \(x\).
• This switch makes the inverse different from the original function, and finding this inverse forms the crux of many algebraic problems.

In the exercise, transforming \(y = \log_{0.5}(x)\) into its inverse involved turning it to the exponential form: \(y = 0.5^x\).
Interchanging \(x\) and \(y\) perfectly outlines the inversion strategy applied here.
Inversion is a powerful tool in understanding how different functions are interrelated and in transforming complex expressions intuitively.