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: $\log x = \log_{10} x$.

Definition

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 $\log 100$.

Answer

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

Laws of Logarithms

Proposition Laws of Logarithms

For $x>0$ and $y>0$, and for any real number $k$:

Product Rule: $\log (xy) = \log x + \log y$
Quotient Rule: $\log \left(\dfrac{x}{y}\right) = \log x - \log y$
Power Rule: $\log (x^k) = k \log x$

Proof

Assume $x>0$ and $y>0$. We use the fact that $x = 10^{\log x}$ and $y = 10^{\log y}$.$$\begin{aligned}\log(xy) &= \log(10^{\log x} \times 10^{\log y}) \\ &= \log\bigl(10^{\log x + \log y}\bigr) \\ &= \log x+ \log y \\ \\ \log\left(\frac{x}{y}\right) &= \log\left(\frac{10^{\log x}}{10^{\log y}}\right) \\ &= \log\bigl(10^{\log x - \log y}\bigr) \\ &= \log x - \log y \\ \\ \log(x^k) &= \log\left(\bigl(10^{\log x}\bigr)^k\right) \\ &= \log\bigl(10^{k \log x}\bigr) \\ &= k \log x\\ \end{aligned}$$

Example

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

Answer

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

Using Logarithms to Solve Exponential Equations

Method Solving Exponential Equations Using Logarithms

To solve an exponential equation of the form $a^x = b$ (where $a > 0$, $a \neq 1$, $b > 0$):

Take the logarithm (base 10) of both sides: $\log(a^x) = \log b$
Apply the power rule: $x \log a = \log b$
Solve for $x$: $x = \dfrac{\log b}{\log a}$

(You could use any logarithm base; using base 10 matches the $\log$ key on most calculators.)

Example

Solve $2^x = 7$.

Answer

$$\begin{aligned}2^x &= 7 \\ \log(2^x) &= \log 7 &&\text{(taking log of both sides)}\\ x \log 2 &= \log 7 &&\text{(power rule)}\\ x &= \frac{\log 7}{\log 2} &&\text{(dividing both sides by } \log 2\text{)}\\ x &\approx 2.807 &&\text{(using calculator)}\\ \end{aligned}$$

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 $[H^+]$.

Answer

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