1.2 Powers, roots and logarithms

Powers and roots are taught early in school because they are basic tools for later mathematics. They belong together: a power builds a result by repeated multiplication, while a root can recover the starting number. A logarithm asks a related question: which exponent produced the result? These three questions appear again in algebra, growth models, and data scales.

For example, the same relationship can be read in three ways:

Powers: repeated multiplication

In $a^n$, $a$ is the base and $n$ is the exponent. For a positive whole-number exponent, $a^n$ means multiplying $n$ copies of $a$: $2^4=2\times2\times2\times2=16$. The exponent counts the copies; $2^4$ is not $2\times4$.

A quick tour of exponent rules

For the following rules, take $a,b>0$ and let $m,n$ be real numbers:

The base must match before you combine exponents by addition or subtraction. Also, adding two powers is different from multiplying them: $2^3+2^2=8+4=12$, while $2^{3+2}=32$.

Roots: finding a starting number

If a square has area $9$ square units, each side is $3$ units long because $3^2=9$. This is what $\sqrt{9}=3$ tells us.

The symbol $\sqrt{x}$ means the nonnegative square root. So $\sqrt{9}=3$, even though both $3^2$ and $(-3)^2$ equal $9$. If you solve the equation $x^2=9$, you must include both solutions, $x=3$ and $x=-3$. More generally, $\sqrt{a^2}=|a|$, the distance of $a$ from zero (see the next module).

Roots can also be written as fractional exponents. For $a>0$, a positive whole number $n$ and an integer $m$,

$$\sqrt[n]{a}=a^{1/n},\qquad a^{m/n}=\left(\sqrt[n]{a}\right)^m.$$

For example, $\sqrt{16}=16^{1/2}=4$ and $8^{2/3}=(\sqrt[3]{8})^2=2^2=4$. A cube root can also take a negative number: $\sqrt[3]{-8}=-2$. For nonnegative $a$ and $b$, $\sqrt{ab}=\sqrt a\sqrt b$; if $b>0$, then $\sqrt{a/b}=\sqrt a/\sqrt b$. The product rule is useful when a square number is hidden inside a larger root, such as $\sqrt{72}=\sqrt{36\cdot2}=6\sqrt2$.

Logarithms: finding an exponent

Suppose $2^x=8$. The unknown is now the exponent. A logarithm names that exponent: $x=\log_2 8=3$. In general,

$$\log_b x=y\quad\Longleftrightarrow\quad b^y=x, \qquad b>0,\ b\ne1,\ x>0.$$

The base $b$ is written as a subscript after $\log$. You can check a logarithm by converting it back to a power. For example, $\log_{10}1000=3$ because $10^3=1000$. The notation $\ln x$ means a logarithm with base $e\approx2.718$.

The graph shows the same connection. Every point $(u,v)$ on $y=2^x$ becomes the point $(v,u)$ on $y=\log_2 x$. For instance, $(1,2)$ on the first graph corresponds to $(2,1)$ on the second.

A quick tour of logarithm rules

For $x,y>0$ and $b>0$ with $b\ne1$:

$$\log_b(xy)=\log_b x+\log_b y,\qquad \log_b\!\left(\frac{x}{y}\right)=\log_b x-\log_b y,\qquad \log_b(x^k)=k\log_b x.$$

These follow from the exponent rules: multiplying two powers of $b$ adds their exponents. You also have $\log_b1=0$, $\log_b b=1$, $\log_b(b^t)=t$, and $b^{\log_b x}=x$. To change the base, use $\log_b x=\ln x/\ln b$. The product rule applies to multiplication, not addition: $\log_b(x+y)$ is generally not $\log_b x+\log_b y$.

Application 1: a square that grows

A square has area $2^8$ square centimetres. Its side length is doubled. What is the original side length, and what is the new area?

1. Recover the side from the area. A side squared gives the area, so the original side is $\sqrt{2^8}=(2^8)^{1/2}=2^4=16$ cm.
2. Double the side. The new side is $2\cdot16=32=2^5$ cm.
3. Square the new side. The new area is $(2^5)^2=2^{10}=1024$ square centimetres.

Check it another way: doubling both dimensions makes the area $2^2=4$ times as large. The original area was $2^8=256$, and $4\cdot256=1024$. A root found the missing length; exponent rules made the new area quick to calculate.

Application 2: how many doublings?

A dataset starts with $30$ records and doubles after every round. After how many complete rounds will it first contain at least $1000$ records?

1. Write the growth rule. After $n$ rounds, the dataset has $30\cdot2^n$ records. We need $30\cdot2^n\ge1000$.
2. Isolate the power. Divide by $30$: $2^n\ge1000/30$.
3. Find the exponent. Since base $2$ is greater than $1$, taking $\log_2$ keeps the inequality's direction: $n\ge\log_2(1000/30)\approx5.06$. The first whole number that works is $n=6$.
4. Check the boundary. After $5$ rounds there are $30\cdot2^5=960$ records, still below $1000$. After $6$ rounds there are $30\cdot2^6=1920$.

Here the logarithm answers an exponent question, while the final check turns that continuous answer into a number of complete rounds.