Problem 66 - Algebra 2

Question

Find the inverse of each function. $$ y=\log _{10} x $$

Step-by-Step Solution

Answer

The inverse of the function \(y=\log _{10} x\) is \(f^{-1}(x)=10^x\)

Step 1: Identify the original function

This step is pretty simple as the original function is already given: \(y=\log _{10} x\)

Step 2: Switch the roles of x and y

Replace every occurrence of \(x\) with \(y\) and \(y\) with \(x\) to get: \(x=\log _{10} y\)

Step 3: Convert the logarithmic equation to an exponential equation

In this step, we need to convert the logarithmic equation to an exponential equation. The rule states that if \(a=\log_{b}c\), then \(b^a=c\). So applying this to our equation, we get: \(10^x=y\)

Step 4: The Inverse Function

Now let's name the inverse function \(f^{-1}(x)\). Therefore, the inverse of the original function is: \(f^{-1}(x)=10^x\)

Key Concepts

Logarithmic Functions

Logarithmic functions are a unique type of mathematical function used to describe the relationship between quantities undergoing exponential growth or decay. The logarithmic function, often written as \(y = \log_{b} x\), tells you what power you need to raise the base \(b\) to obtain the number \(x\). It is essentially the inverse of an exponential function, which makes it crucial for solving problems involving exponential change.

Logarithms have a wide range of applications including in the fields of science, engineering, finance, and even music. One of their most common forms is the natural logarithm, where the base \(b\) is the constant \((e=2.718)\). However, in this exercise, we encountered the logarithm with a base of 10, often referred to as the common logarithm.

• They convert multiplication into addition, which simplifies complex calculations.
• Logarithms are used to measure scales such as the Richter scale for earthquakes and the pH level in chemistry.

Understanding logarithmic functions requires recognizing these notations and their rules, which then simplify when solving their inverses.

Exponential Functions

Exponential functions are mathematical functions of the form \(f(x) = a^x\), where \(a\) is a positive real number, and \(x\) can be any real number. These functions describe processes that grow or decay at a rate proportional to their current value. Unlike linear growth, exponential growth accelerates over time.

In this problem, after swapping and transforming the original function from logarithmic to exponential, we obtained \(y = 10^x\), an example of an exponential function. It shows how the variable \(x\) represents the power to which the base (10 in this case) is raised.

• Exponential functions are used in population growth models, radioactive decay, and compounding interest in finance.
• The base \(a\) determines the growth rate: if \(a > 1\), it is growth; if \(0 < a < 1\), it is decay.

Learning exponential functions is key to understanding natural processes and many complex systems in our world.

Function Transformation

Function transformation involves adjusting a function's formula to achieve various effects on its graph. These transformations typically include translations, reflections, stretching, and compressing.

In the exercise, the transformation occurs when we find the inverse function. By switching from the original logarithmic equation to the exponential form, we effectively transform the function, illustrating an inversion of roles between \(x\) and \(y\).

• Translating a function shifts its graph up/down or left/right.
• Reflecting can flip the graph over an axis.
• Stretching or compressing modifies the steepness or shallowness of the graph.

Learning function transformations helps visualize how equations relate to the graphical representations of functions, enhancing one's ability to predict and understand behavior changes.