What is a Logarithm Function?

A logarithm is a tool that helps us answer a question:

“What exponent do I need to raise a specific base to, in order to get a certain number?”

The Form

The “log” is the exponent. The following two statements mean the exact same thing:

\[\log_b(c) = x \quad \iff \quad b^x = c\]
  • \(b\) is the base.
  • \(x\) is the exponent (the answer to the log).
  • \(c\) is the result.

The Infamous Base 10

Let’s use the base we all know, \(b=10\).

  • We know \(10^2 = 100\).

  • Therefore, \(\log_{10}(100) = 2\).

  • We know \(10^0 = 1\).
  • Therefore, \(\log_{10}(1) = 0\).

A log is just the exponent.

The Natural Logarithm (\(\ln\))

From our dive into the Derivation of an Exponential Function, we discovered the special base \(e \approx 2.718...\).

A logarithm with base $e$ is called the Natural Logarithm and is written as \(\ln(x)\).

\[\ln(x) \quad \text{is just shorthand for} \quad \log_e(x)\]

It asks the question: “What power do I raise \(e\) to, to get \(x\)?”

Applying Logarithms to Functions

Let’s see what happens when we take the natural log of our two function types, using the Logarithm Power Rule.

1. Power Function: \(y = x^2\)

If we take the log of both sides:

\[\ln(y) = \ln(x^2)\]

Using the Power Rule (\(\ln(A^B) = B \cdot \ln(A)\)), this becomes:

\[\ln(y) = 2 \cdot \ln(x)\]

2. Exponential Function: \(y = e^x\)

If we take the log of both sides:

\[\ln(y) = \ln(e^x)\]

Using the Power Rule, this becomes:

\[\ln(y) = x \cdot \ln(e)\]

And since \(\ln(e)\) (which is \(\log_e(e)\)) is just 1, this simplifies to:

\[\ln(y) = x\]

The Conclusion

The logarithm of a power function (\(\ln(y) = 2 \cdot \ln(x)\)) doesn’t free the \(x\). It’s still “trapped” inside another function.

However, the logarithm of an exponential function (\(\ln(y) = x\)) frees the \(x\) completely. The \(x\) is now isolated.

To sum up: when working with functions, the place where the independent variable exists matters.

  1. For exponential functions (\(e^x\)), the log is the inverse (it solves for the exponent).
  2. For power functions (\(x^2\)), the log is a simplifier (it turns powers into multiplication).