The logarithm function is an important function that arises from finding the inverse function of an exponential. More specifically,

if \( a^y = x \), then \( y = \log_a(x) \).

Definition

The logarithm of \( x \) to base \( a \) is:

\[\begin{align*} \log_a(x) = y \quad \text{if and only if} \quad a^y = x, \end{align*}\]

where \( a > 0 \), \( a \neq 1 \), and \( x > 0 \).

Inverse Function

The logarithm is the inverse function of the exponential function. For \( y = a^x \), the inverse is \( x = \log_a(y) \).

Special Case: Natural Logarithm

When \( a = e \), where \( e \approx 2.718 \), we denote \( \log_e(x) \) as \( \ln(x) \), known as the natural logarithm:

\[\begin{align*} \ln(x) = y \quad \text{if and only if} \quad e^y = x. \end{align*}\]

Examples and Applications

Logarithms are widely used in various fields:

• Finance: Logarithms and exponential growth are often found when discussing exponential growth.
• Physical Sciences: Many scales in the physical sciences, such as the pH scale for measuring a substance’s acidity/alkalinity or the decibel scale for measuring sound intensity, are logarithmic scales.
• Computer Science: Logarithms are often used to describe time and space complexity in computer science, such as \( O(\log n) \). These describe how efficiently algorithms generalize to larger inputs.

Practice Problems and Solutions

Problem 1: Simplify \( \log_2(8) \).

Solution: Since \( 2^3 = 8 \), we have:

\[\begin{align*} \log_2(8) = 3. \end{align*}\]

Problem 2: Evaluate \( \ln(e^5) \).

Solution: Since the logarithm is the inverse of the exponential function, \( \ln(e^5) = x \) can be rewritten as

\[\begin{align*} e^x = e^5, \end{align*}\]

so $x = 5$, or $\ln(e^5) = 5.$

Problem 3: Evaluate \( \log_3(81) \).

Solution: Since \( 3^4 = 81 \), we have:

\[\begin{align*} \log_3(81) = 4. \end{align*}\]

Thus, the answer is \( 4 \).