Logarithms as an Inverse Function

An inverse function \(f^{-1}(x)\) is a function that “undoes” the original function \(f(x)\).

If we have a function \(y = f(x)\), its inverse \(f^{-1}(y)\) takes the output \(y\) and “spits out” the original input \(x\) .

We can find the formula for the inverse by taking our original function \(y = f(x)\) and algebraically solving for \(x\) .

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Example 1: A Simple Function

Let’s find the inverse of \(f(x) = x + 5\).

1. Start with the function: If the result of \(f(x)\) is \(y\), then:

   \[y = x + 5\]

2. Solve for \(x\): (We rearrange the equation to isolate \(x\))

   \[x = y - 5\]

3. This is the inverse: The formula that gives us \(x\) is our inverse function. We know \(f^{-1}(y) = x\). Therefore:

   \(f^{-1}(y) = y - 5\) (Or, written in terms of \(x\): \(f^{-1}(x) = x - 5\))

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Example 2: The Exponential Function

Now, let’s find the inverse of \(f(x) = e^x\).

1. Start with the function:

   \[y = e^x\]

2. Solve for \(x\): The \(x\) is “trapped” in the exponent. To get it out, we must apply the only tool that can “undo” an \(e\) exponent: the natural logarithm (\(\ln\)).

   We take the \(\ln\) of both sides:

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

   Now, we use the Logarithm Power Rule (\(\ln(A^B) = B \cdot \ln(A)\)) to bring the \(x\) down:

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

   And since we know \(\ln(e) = 1\):

   \[\ln(y) = x \cdot 1\] \[x = \ln(y)\]

3. This is the inverse: The formula that gives us \(x\) is the natural logarithm.

   \(f^{-1}(y) = \ln(y)\) (Or, written in terms of \(x\): \(f^{-1}(x) = \ln(x)\))

This proves that the Natural Logarithm is the inverse function of the Exponential Function \(e^x\).