Oftentimes in science and economics we make use of very large numbers or perhaps very small numbers. Numbers such as $34~927~000~000~000~000$ or perhaps $0.0000000012$ can be cumbersome to write. Instead of worrying about how many zeros there are to the right of the decimal with very small numbers, or how many trailing zeros there are with very large numbers, we write these numbers in scientific notation which are numbers of the form:

$a \times 10b$ where $1 \leq a < 10$ and $a\in\mathbb R$ and $b \in \mathbb Z$

Simply put, $a$ is a real number between $1$ and $10$ and $b$ is an integer. We are often tasked with multiplying or dividing these very small or very large numbers with each other. Rules have been established regarding operations with exponents. These rules can be considered very useful shortcuts and should be utilized in the A&I SL curriculum. They are as follows:

1. $xa \cdot xb = x^{a+b}$
2. $\dfrac{x^a}{x^b}=x^{a-b}$
3. $(x^a)^b = x^{ab}$
4. $x^{-a} = \dfrac{1}{x^a}$

Example 1: Express $170~000~000~000~000$ and $0.000000042$ in scientific notation and express their product in scientific notation.

Explanation: $170~000~000~000~000 = 1.7 \times 10^{14}$ and $0.000 000 042 = 4.2 \times 10^{−8}$.

Multiplying these numbers involves multiplying the coefficients and adding the exponents per the rules for exponents stated previously.

$(1.7 \times 10^{14})(4.2 \times 10^{−8}) = 7.14 \times 10^6$

Example 2: Simplify $\overset{3_4 \cdot 3_{−7}}{\_\_\_}(3_2)_{12}$ and express your answer using positive exponents.

Explanation: $\dfrac{3_4 \cdot 3_{−7}}{(3^2)^{12}}$ $= \dfrac{3^{-3}}{3^{24}}$ $= 3^{-3-24}$ $= 3^{-27}$ $= \dfrac{1}{3^{27}}$

The exponential equivalent of an equation in logarithmic form is: If $10a = b$, then $\log b = a$

Example 3: Determine the value of $\log 38$.

Here we are asked to determine what power $10$ must be raised to in order to obtain $38$.

Using the TI-84 CE graphing calculator gives:

Thus, to $3$ significant figures, the solution is $1.58$ because $10^{1.58} = 38$.