Lecture #15: Logarithms: Properties, Applications & Derivatives

MATH 145 – Calculus I

Torin Quinlivan
tquinlivan@knox.edu

Knox College

October 16, 2025

Logarithmic Functions

We know that if: \[ x = 2^3, \] then \(x = 8\).

But how would we solve the following equation for \(x\)? \[ 2^x = 8 \]

Logarithmic Functions

Given an exponential function \(b^x\), the logarithm to the base \(b\) is the function denoted by \(\log_b(x)\) and defined by the equations \[ \begin{aligned} \log_b(b^x) &= x & &\textrm{and} & b^{\log_b(x)} = x \end{aligned} \]

The logarithm is the inverse operation of exponentiation, just like how division is the inverse operation of multiplication or how subtraction is the inverse operation of addition.

Common Logarithm, \(\log_10(x)\)

\(\log_10(x)\) is called the common logarithm and is generally denoted by \(log(x)\).

Base 10 is the numeral system we use in general life.

Natural Logarithm

\(\log_e(x)\) is called the natural logarithm and is typically denoted by \(\ln(x)\).

Note that: \[ \begin{aligned} \ln(e^x) &= x & &\textrm{and} & e^{\ln(x)} &= x \end{aligned} \]

The natural logarithm is the most important logarithm and the one you will use most often.

Caution! Because it’s so important and widely-used, occasionally you may see online sources use to refer to and not . Make sure you read what they are doing closely. I will try my best to always use the notation described in this lecture.

Example: Logarithms

Examples:

  1. Solve \(e^{7x} = 12\).
  2. Solve \(4 + 2e^{-x} = 8\).
  3. Solve \(10000^x + 75 = 98\).

Properties of Logarithms

Note that \(\log(x)\) and \(\ln(x)\) are not defined when \(x\) is negative or \(0\).

  1. \(\log(ab) = \log(a) + \log(b)\).
  2. \(\log\left(\dfrac{a}{b}\right) = \log(a) - \log(b)\).
  3. \(\log(a^p) = p \log(a)\).
  4. \(\log(10^x) = x\).
  5. \(10^{\log(x)} = x\)
  1. \(\ln(ab) = \ln(a) + \ln(b)\).
  2. \(\ln\left(\dfrac{a}{b}\right) = \ln(a) - \ln(b)\).
  3. \(\ln(a^p) = p \ln(a)\).
  4. \(\ln(e^x) = x\).
  5. \(e^{\ln(x)} = x\)

\(\log_b(x) = \dfrac{\ln(x)}{\ln(b)}\). (This works for any base, not just “\(\ln\)”.)

Example: Logarithms

Examples:

  1. Solve \(4(2^t) - 2 = 3\).
  2. Solve \(3 + 2 \log(t) = 3.5\).
  3. Solve \(5 = 4(2^{x - 1})\).

Example: Logarithms

Example: One hundred kilograms of radioactive substance decays to \(40 kg\) in \(10\) years.

  1. How much material is left after 20 years?
  2. What is the half-life of the material? How long until \(50 kg\) of the material remains?

Derivative of the Natural Logarithm

Recall: \(\dfrac{d}{dx} e^x = e^x\), and by the Chain Rule, \(\dfrac{d}{dx} e^{f(x)} = e^{f(x)} f'(x)\).

What is \(\dfrac{d}{dx} \ln(x)\)?

Derivative of the Natural Logarithm

Consider \(x = e^{\ln(x)}\). \[\begin{align*} x &= e^{\ln(x)} \\ \frac{d}{dx} x &= \frac{d}{dx} e^{\ln(x)} \\ 1 &= e^{\ln(x)} \cdot \left( \frac{d}{dx} \ln(x) \right) \\ 1 &= x \left( \frac{d}{dx} \ln(x) \right) \\ \frac{d}{dx} \ln(x) &= \frac{1}{x} \end{align*}\]

Derivative of the Natural Logarithm

\[ \frac{d}{dx} \ln(x) = \frac{1}{x} \]

In general: \[ \begin{aligned} \frac{d}{dx} \ln(f(x)) = \frac{f'(x)}{f(x)} \end{aligned} \]

Example: Derivative of the Natural Logarithm

Examples:

  1. Solve \(\dfrac{d}{dt} \ln(3t + 1)\).
  2. Solve \(\dfrac{d}{dx} x^2 \ln(x^2 + 2)\).
  3. Solve \(\dfrac{d}{dx} \ln(2^{3x + 4})\).

Derivative of the Natural Logarithm: Consequences

\[ \frac{d}{dx} a^x = a^x \ln(a) \]

\[ \frac{d}{dx} \log_a(x) = \frac{1}{x \ln(a)} \]

Example: Derivative of the Natural Logarithm

Examples:

  1. Solve \(\dfrac{d}{dt} 2^{3x}\).
  2. Solve \(\dfrac{d}{dx} x^2 \log(3x)\).