Question

Find the inverse of the function. $$ y=10^{x-3} $$

Short answer

The inverse of the function \(y = 10^{x-3}\) is \(y = \log_{10}{x} + 3\).

Step-by-step solution

Swap x and y

Remember, the inverse of a function can be found by swapping x and y of the original function. So, rewrite \(y = 10^{x-3}\) as \(x = 10^{y-3}\)

Take logarithm of both sides

To bring \(y - 3\) down from the exponent, we take the logarithm of both sides. So apply the logarithm to both sides of the equation: \(\log_{10}{x} = \log_{10}{10^{y-3}}\).

Simplify

In logarithmic form, an exponent can be brought down and be multiplied at the front of the logarithm. The equation \(\log_{10}{x} = \log_{10}{10^{y-3}}\) becomes \(\log_{10}{x} = (y-3)\log_{10}{10}\). Because \(\log_{10}{10} = 1\), we simplify this to \(\log_{10}{x} = y - 3\).

Isolate y

Finally, to get the inverse function, solve the equation for \(y\). Add 3 to both sides of the equation: \(y = \log_{10}{x} + 3\).

Key concepts

Logarithmic Functions

Logarithmic functions are the inverse operations of exponential functions. To understand this, let's first recall that an exponential function takes the form of \( y = b^x \), where \( b \) is the base and \( x \) is the exponent. Now, a logarithmic function would reverse this operation, represented as \( x = \log_b(y) \), which essentially asks the question: to what power must we raise \( b \) to get \( y \)? So, if we take our example from the exercise, \( y = 10^{x-3} \), and we want to express \( x \) in terms of \( y \), a logarithmic function can help us do that.

When finding the inverse of an exponential function, taking the logarithm of both sides, as done in Step 2 of the solution, allows us to handle the exponent (in our case \( x - 3 \)) as a multiplicative factor. That's why logarithms are crucial for dealing with exponential equations, especially when it comes to finding their inverses.

Exponential Functions

Exponential functions feature a constant base raised to a variable exponent. They have the general form \( y = a^x \), where \( a \) is a positive real number, and \( x \) is the exponent. In our textbook problem, \( y = 10^{x-3} \) is an exponential function with base 10. Exponential functions are unique in their growth rate; they increase (or decrease if the exponent is negative) at a rate proportional to their current value, which makes them show up frequently in the realms of compounding interest, population growth, and decay processes among others.

An exponential function's graph is distinctive for its rapid increase or decrease and never touching the x-axis, reflecting the idea that no matter what power you raise a positive number to, the result will never be zero. Understanding the nature of exponential functions is key when it comes to finding inverses since their inverses will exhibit this growth pattern in a logarithmic form.

Algebraic Manipulation

Algebraic manipulation is a set of tools and techniques used to rearrange algebraic expressions and solve equations. It's all about the manipulation of equations to make unknown variables the subject of the formula. This process often involves operations such as addition, subtraction, multiplication, division, factoring, expanding, and applying inverse operations.

In terms of our exercise, after swapping \( x \) and \( y \), and taking the logarithm of both sides, we reach a pivotal moment of algebraic manipulation. The equation \( \log_{10}{x} = (y-3)\log_{10}{10} \) is simplified because the logarithm of a number to its own base, \( \log_{b}{b} \), equals one. This step showcases how understanding the properties of logarithms can aid in algebraic manipulation to simplify expressions and ultimately isolate the variable of interest, which in this case is \( y \).

Inverse Function Properties

Understanding the properties of inverse functions is essential when finding the inverse. One of the key properties is that the graph of an inverse function is a reflection of the original function's graph over the line \( y = x \). This reflection correlates with the algebraic step of swapping \( x \) and \( y \) as seen in Step 1 of the solution.

Another important property is that applying an inverse function to a value reverses the effect of the original function. If you have an exponential function and apply its logarithmic inverse, you essentially 'undo' the exponential operation. This is evident in our Step 3, where applying the logarithm simplifies the equation to a point where we can isolate \( y \) easily. When both the function and its inverse are placed together, say \( f(x) \) and \( f^{-1}(x) \), and one is applied after the other, the result will be the original value \( x \). This property can also serve as a check to ensure the inverse has been found correctly; plugging values into both the function and its inverse should yield the original input.