Exponential and Logarithmic Functions: Inverse Relationships

The Exponential Function

Let's begin with the exponential function, which you've likely encountered before. It is characterized by having a variable in the exponent.

An exponential function is a function of the form:

where the base $b$ is a positive constant, and $b\ne 1$. The behavior of this function changes dramatically based on the value of $b$.

The following resource provides a clear, visual introduction to the graphs and properties of exponential functions.

As you saw, the key characteristics of the exponential function $f(x)=b^x$ are:

• Domain: All real numbers, $(-\infty ,\infty )$.
• Range: All positive real numbers, $(0,\infty )$.
• Horizontal Asymptote: The x-axis ($y=0$). The graph gets infinitely close to this line but never touches it.
• Y-intercept: The graph always passes through the point $(0,1)$, since $b^0=1$ for any base $b$.
• Key Point: The graph also passes through the point $(1,b)$.
• Behavior:
  • If $b>1$, the function is increasing and exhibits exponential growth.
  • If $0<b<1$, the function is decreasing and exhibits exponential decay.

The natural exponential function, $f(x)=e^x$, where $e\approx 2.71828$, is the most important exponential function in calculus. Since $e>1$, its graph shows exponential growth.

Logarithms as the Inverse of Exponentials

If you apply the horizontal line test to the graph of any exponential function, you'll see that it passes. This means that exponential functions are one-to-one, and therefore, they must have an inverse function. This inverse is called the logarithmic function.

Professor Leonard provides an outstanding conceptual explanation of where logarithms come from and how they are fundamentally linked to exponentials.

The video establishes the central idea: the logarithm is the function that "undoes" the exponential. This leads to the fundamental definition that connects the two:

This equivalence is key. It tells us that a logarithm is an exponent. The expression ${\log }_{b}y$ asks the question: "To what power must we raise the base $b$ to get the value $y$?"

Two special logarithms are essential:

1. Common Logarithm: Base 10, written as $\log (x)$. The base is implied.
2. Natural Logarithm: Base $e$, written as $\ln (x)$. This is the inverse of the natural exponential function $e^x$.

The Logarithmic Function

Now, let's formalize the properties of the logarithmic function. Since it is the inverse of the exponential function, its properties are a direct reflection of what we've already learned.

Let's summarize the key characteristics of the logarithmic function $f(x)={\log }_{b}x$, which are the "mirror image" of the exponential function's properties:

• Domain: All positive real numbers, $(0,\infty )$. This is crucial; you cannot take the logarithm of a negative number or zero.
• Range: All real numbers, $(-\infty ,\infty )$.
• Vertical Asymptote: The y-axis ($x=0$).
• X-intercept: The graph always passes through the point $(1,0)$.
• Key Point: The graph also passes through the point $(b,1)$.
• Behavior:
  • If $b>1$, the function is increasing.
  • If $0<b<1$, the function is decreasing.

The inverse relationship is perfectly captured by reflecting the graph of an exponential function across the line $y=x$.

Conclusion

In this lesson, we have added the exponential and logarithmic functions to your calculus toolkit. We've seen that they are not just two separate topics, but two sides of the same coin, linked by the concept of an inverse.

Key Takeaways:

• Exponential functions ($f(x)=b^x$) are defined by their base $b$. They have a domain of all real numbers, a range of positive numbers, and a horizontal asymptote at $y=0$. They model rapid growth ($b>1$) or decay ($0<b<1$).
• Logarithmic functions ($f(x)={\log }_{b}x$) are the inverses of exponential functions. The expression ${\log }_{b}x$ represents the exponent to which $b$ must be raised to obtain $x$.
• The Inverse Relationship: All properties of logarithmic functions are reflections of their exponential counterparts. The domain and range are swapped, key points are reversed (e.g., $(0,1)$ becomes $(1,0)$), and the horizontal asymptote becomes a vertical asymptote. Graphically, they are mirror images across the line $y=x$.

In our next lesson, we will build upon this foundation. We will address the next learning outcome: Graph complex functions by applying horizontal and vertical shifts, stretches, and reflections to parent functions. The functions we've studied today, like $e^x$ and $\ln x$, will serve as our new "parent functions" for these transformations.