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Logarithms

In the previous chapter, we explored exponentiation, which answers the question of what we get when we multiply a number by itself a certain number of times (for example, \(10^3 = 10 \times 10 \times 10 = 1000\)).
Now we will ask the inverse question: To what exponent must 10 be raised to get 1000?
This question leads us to the concept of the logarithm. A logarithm is the inverse operation of exponentiation: it is the tool we use to find an unknown exponent.
Before the invention of calculators, logarithms were a revolutionary tool for scientists, turning complex multiplications into simpler additions. Today, they remain essential for solving exponential equations and are used to describe phenomena in science, such as the pH scale in chemistry or the Richter scale for earthquakes.
In this chapter, we will:
  • define logarithms (in base 10),
  • explore their main properties (the laws of logarithms),
  • and use them to solve exponential equations and real-world problems.
Throughout this chapter, unless stated otherwise, we use \(\log x\) to mean the logarithm base 10: \(\log x = \log_{10} x\).

Logarithms in Base 10

Definition Logarithm
The (base 10) logarithm of a number \(y\) (where \(y > 0\)) is the exponent to which 10 must be raised to obtain \(y\). It is denoted as:$$\log y = x \quad \Leftrightarrow \quad 10^x = y$$In other words, \(\log y\) is the exponent \(x\) such that \(10^x = y\).
Proposition Inverse Properties of Logarithms
For real numbers:
  • \(10^{\log x} = x\) for any \(x > 0\),
  • \(\log (10^x) = x\) for any real \(x\).
These properties reflect that the exponential function \(10^x\) and the logarithm function \(\log x\) (base 10) are inverses of each other.
Example
Evaluate \(\log 100\).

\(\begin{aligned}[t]\log(100) &= \log\left(10^{2}\right) \\&=2\\\end{aligned}\)
Since \(10^2 = 100\), the logarithm of \(100\) (base 10) is \(2\).

Logarithms in Base \(a\)

In the previous section we defined the logarithm in base \(10\) of a number as the exponent to which \(10\) must be raised in order to obtain that number.
We can use the same principle to define logarithms in other bases:
Definition Logarithm in Base a
For a positive number \(a \neq 1\), the logarithm base \(a\) of a number \(y\) (where \(y > 0\)) is the exponent to which \(a\) must be raised to obtain \(y\). It is denoted as:$$\log_a y = x \quad \Leftrightarrow \quad a^x = y.$$
Proposition Inverse Properties of Logarithms in Base a
Let \(a>0\) with \(a\neq 1\).
  • \(a^{\log_a x} = x\) for any \(x > 0\),
  • \(\log_a (a^x) = x\) for any real \(x\).
These properties show that the exponential function \(a^x\) and the logarithm function \(\log_a x\) are inverses of each other.
Example
Evaluate \(\log_2 8\).

\(\begin{aligned}[t]\log_2 8 &= \log_2 (2^3) \\&=3\\\end{aligned}\)

Proposition Laws of Logarithms in Base a
For \(x>0\), \(y>0\), and \(a > 0\), \(a \neq 1\):
  • Product Rule: \(\log_a (xy) = \log_a x + \log_a y\)
  • Quotient Rule: \(\log_a \left(\dfrac{x}{y}\right) = \log_a x - \log_a y\)
  • Power Rule: \(\log_a (x^k) = k \log_a x\)

$$\begin{aligned}\log_a(xy) &= \log_a\bigl(a^{\log_a x} \times a^{\log_a y}\bigr) \\ &= \log_a\bigl(a^{\log_a x + \log_a y}\bigr) \\ &= \log_a x + \log_a y \\ \\ \log_a\left(\frac{x}{y}\right) &= \log_a\left(\frac{a^{\log_a x}}{a^{\log_a y}}\right) \\ &= \log_a\bigl(a^{\log_a x - \log_a y}\bigr) \\ &= \log_a x - \log_a y \\ \\ \log_a(x^k) &= \log_a\left(\bigl(a^{\log_a x}\bigr)^k\right) \\ &= \log_a\bigl(a^{k \log_a x}\bigr) \\ &= k \log_a x\\ \end{aligned}$$

Method Solving Exponential Equations with Base a
To solve \(a^x = b\) (where \(a > 0\), \(a \neq 1\), \(b > 0\)):
  1. Take the logarithm base \(a\) of both sides: \(\log_a (a^x) = \log_a b\)
  2. Simplify: \(x = \log_a b\)
Example
Solve \(3^x = 81\).

$$\begin{aligned}3^x &= 81 \\ \log_3 (3^x) &= \log_3 (81) \\ x &= \log_3 (3^4) \\ x &= 4\end{aligned}$$

Natural Logarithm

The natural logarithm, often denoted \(\ln x\), is the logarithm with base \(e\) (where \(e \approx 2.71828\) is the base of the natural exponential function). It is the inverse of the exponential function \(e^x\) and plays a central role in calculus, growth models, and many scientific applications.
Definition Natural Logarithm
The natural logarithm, denoted \(\ln x\), is the logarithm base \(e\), defined for \(x>0\):$$\ln x = \log_e(x).$$
Proposition Inverse Properties of Natural Logarithms
  • \(e^{\ln x} = x\) for \(x > 0\),
  • \(\ln (e^x) = x\) for any real \(x\).

Laws of Logarithms

Proposition Laws of Logarithms
For \(x>0\), \(y>0\) and \(a>0\) with \(a\neq 1\):
  • Product Rule: \(\log_a (xy) = \log _a x + \log _a y\)
  • Quotient Rule: \(\log _a \left(\dfrac{x}{y}\right) = \log _a x - \log_a y\)
  • Power Rule: \(\log_a (x^k) = k \log_a x\)

$$\begin{aligned}\log_a(xy) &= \log_a\bigl(a^{\log_a x} \times a^{\log_a y}\bigr) \\ &= \log_a\bigl(a^{\log_a x + \log_a y}\bigr) \\ &= \log_a x+ \log_a y \\ \\ \log_a\left(\frac{x}{y}\right) &= \log_a\left(\frac{a^{\log_a x}}{a^{\log_a y}}\right) \\ &= \log_a\bigl(a^{\log_a x - \log_a y}\bigr) \\ &= \log_a x - \log_a y \\ \\ \log_a(x^k) &= \log_a\left(\left(a^{\log_a x}\right)^k\right) \\ &= \log_a\bigl(a^{k \log_a x}\bigr) \\ &= k \log_a x\\ \end{aligned}$$

Example
Write as a single logarithm: \(\log (5)+\log (3)\).

\(\begin{aligned}[t]\log (5)+\log (3) &= \log (5\times 3) \\&= \log 15\\\end{aligned}\)

In particular, for the natural logarithm, for \(x>0\) and \(y>0\):
  • \(\ln (xy) = \ln x + \ln y\)
  • \(\ln \left(\dfrac{x}{y}\right) = \ln x - \ln y\)
  • \(\ln (x^k) = k \ln x\)
Example
Evaluate \(\ln \left(e^4\right)\).

\(\begin{aligned}[t]\ln\left(e^4\right) &= 4 \ln e \\&=4 \times 1 \\&=4\\\end{aligned}\)

Change of Base Rule

Proposition Change of Base Rule
For any \(a>0\), \(a \neq 1\), \(b>0\), \(b \neq 1\), and \(x > 0\):$$\log_a x = \frac{\log_b x}{\log_b a}.$$

Let \(y = \log_a x\).$$ \begin{aligned}y &= \log_a x\\ a^y &= a^{\log_a x}\\ a^y &= x\\ \log_b a^y &= \log_b x \\ y \log_b a &= \log_b x \quad(\text{power rule}) \\ y &= \frac{ \log_b x}{\log_b a} \\ \log_a x &= \frac{ \log_b x}{\log_b a}\quad(\text{as } y = \log_a x). \\ \end{aligned} $$

This rule allows us to compute logarithms in any base using a calculator that computes base \(10\) logarithms or natural logarithms.
Example
Compute \(\log_2 10\) using natural logarithms.

$$\begin{aligned}\log_2 10 &= \frac{\ln 10}{\ln 2} \\ &\approx \frac{2.302585}{0.693147} \\ &\approx 3.321928\end{aligned}$$

Method Using Change of Base
To compute \(\log_a b\) using base \(10\) logarithms:$$\log_a b = \frac{\log b}{\log a}.$$Using natural logarithms:$$\log_a b = \frac{\ln b}{\ln a}.$$

Applications of Logarithms

Logarithms are used to describe quantities that vary over many orders of magnitude. Some important examples in science include:
  • pH scale in chemistry: \(\text{pH} = -\log_{10} [H^+]\)
    (where \([H^+]\) is the hydrogen ion concentration in moles per litre)
  • Richter scale for earthquakes: \(\text{Magnitude} = \log_{10}\left(\dfrac{I}{I_0}\right)\)
    (where \(I\) is the intensity of the earthquake, \(I_0\) is a reference intensity)
  • Decibel (dB) scale for sound: \(L = 10 \log_{10}\left(\dfrac{P}{P_0}\right)\)
    (where \(P\) is the power/intensity measured, \(P_0\) is a reference level)
On calculators, the \(\log\) key usually means \(\log_{10}\), and the \(\ln\) key means the natural logarithm (base \(e\)).
Example
The pH of a solution is \(3.2\). Find the hydrogen ion concentration \([H^+]\).

We know:$$\begin{aligned}\text{pH} &= -\log_{10}[H^+] \\ 3.2 &= -\log_{10}[H^+] &&\text{(substituting the value)}\\ -3.2 &= \log_{10}[H^+] &&\text{(multiplying both sides by \(-1\))}\\ 10^{-3.2} &= 10^{\log_{10}[H^+]} &&\text{(exponentiating both sides, base 10)}\\ [H^+] &= 10^{-3.2} \\ [H^+] &\approx 6.31 \times 10^{-4}\ \text{mol/L} &&\text{(using calculator)}\end{aligned}$$