Why we use this notation

Scientific notation writes any non-zero number in the form

An integer has no fractional part, for example $-3$, $0$ or $12$.

We use this notation to represent the same quantity more compactly, so its size is easier to compare. Distances between planets, microscopic lengths, Avogadro-type counts in chemistry, and global financial figures are much easier to read when the power of $10$ carries the scale.

Examples of large and small values rewritten in scientific notation.

Calculator or computer notation is not acceptable as final mathematical notation. If your calculator shows something like 5.2E30, write it as $5.2 \times 10^{30}$. The letter E is just a calculator shortcut; it is not mathematical multiplication by a power of ten.

Converting into scientific notation

Move the decimal point until exactly one non-zero digit sits to its left. Count how many places it moved: that gives the exponent $k$.

• For a large number, the exponent is positive: $4820000=4.82\times10^6$.
• For a small positive number, the exponent is negative: $0.00073=7.3\times10^{-4}$.

Scientific notation changes how many decimal places you see, but it does not change the number of significant figures. For example, $0.000730$ and $7.30\times10^{-4}$ both show three significant figures.

Operating with scientific notation

For multiplication and division, work with the coefficients and the powers of ten separately, then put the answer back into scientific notation.

$$(3.4\times10^5)(2\times10^7)=6.8\times10^{12}$$

Here we use the exponent law

For division:

$$\frac{8.1\times10^{-3}}{3\times10^4}=2.7\times10^{-7}$$

This uses $\frac{10^m}{10^n}=10^{m-n}$.

For addition and subtraction, first rewrite the numbers with the same power of ten. You can also use your calculator carefully, but make sure the final answer is converted correctly. For example:

$$4.6\times10^8+7.2\times10^7=4.6\times10^8+0.72\times10^8=5.32\times10^8$$

Accuracy and meaning

In IB Mathematics, unless a question says otherwise, final numerical answers should be exact or correct to three significant figures. Exact form means the answer has not been rounded, such as $\frac{13}{7}$, $\sqrt{2}$ or $\pi$. If a question asks for exact form, don't replace it with a rounded decimal.

There is a nice TOK point here: some large numbers have names, while many others are mainly understood through powers of ten. The name we choose affects how comfortably we think about the size. Historically, our present base-ten place-value notation is one stage in a long development of number systems; scientific notation is another representation built on that same place-value idea.

Integer exponent laws

An exponent tells you how many repeated factors of a base are being represented. In $b^n$, $b$ is the base and $n$ is the exponent.

For a non-zero base $b$ and integer exponents $m$ and $n$:

Examples of integer exponent laws and how they simplify.

These laws come from patterns. For example, $5^3\times5^{-6}=5^{-3}$, $6^4\div6^3=6$, and $(2^3)^{-4}=2^{-12}$. If there is a coefficient or a bracket, pause and read the structure carefully: $(2x)^4=16x^4$, but $2x^{-3}=\frac{2}{x^3}$.

The bracket matters. In $(2x)^4$, the whole product $2x$ is raised to the fourth power. In $2x^{-3}$, only the $x$ has exponent $-3$.

Logarithms are exponents

A logarithm is an exponent: it tells you the power to which a positive base must be raised to get a positive number. The basic equivalence is

The conditions are $a>0$, $a\neq1$ and $y>0$.

Keep coming back to this sentence: a logarithm is a power. So $\log_2 8=3$ because $2^3=8$, and $\log_{10}0.001=-3$ because $10^{-3}=0.001$.

A logarithm can be negative, since an exponent can be negative. What you can’t do in this course is take the logarithm of a negative number or use a negative base for the logarithm.

Base $10$ and base $e$

The logarithm to base $10$ is usually written as $\log x$, meaning $\log_{10}x$. The logarithm to base $e$ is written as $\ln x$, meaning $\log_e x$, where $e$ is the mathematical constant approximately equal to $2.718$.

Your calculator evaluates $\log x$ and $\ln x$ directly. Use technology for numerical evaluation, but write the notation properly in your working. For example, write $\log_{10}250$ or $\log250$, not a calculator-button description.

Logarithmic scales are useful when quantities vary over huge ranges. The Richter scale and the decibel scale are common examples; in chemistry, pH calculations also use logarithms. Historically, logarithms made a major difference because they turned difficult multiplication and power problems into easier additive ones, long before electronic calculators existed.

Rational exponents

A rational exponent is an exponent written as a quotient of two integers. Roots and fractional powers are two notations for the same idea.

When $m$ is even, use the positive root. For example, $16^{\frac{3}{4}}=(\sqrt[4]{16})^3=2^3=8$.

More generally,

In real-valued work, watch even roots: the expression under an even root must be non-negative.

The exponent laws from integer powers still guide the algebra, as long as the expressions are defined. For $a>0$, rational exponents simplify just like integer exponents:

$$a^{\frac{1}{2}}a^{\frac{3}{2}}=a^2, \qquad \frac{a^{\frac{5}{3}}}{a^{\frac{2}{3}}}=a$$

Basic logarithm facts

From $y=a^x \Leftrightarrow x=\log_a y$, we get straight away

$$\log_a a=1, \qquad \log_a1=0$$

because $a^1=a$ and $a^0=1$. The base restrictions still apply: $a>0$ and $a\neq1$. The input to the logarithm must be positive as well.

Exponential and logarithmic forms describe the same relationship in inverse ways. That is why logarithmic and exponential graphs, when studied as functions, are related by reflection in the line $y=x$.

Laws of logarithms

The laws of logarithms are algebraic rules that turn multiplication, division and powers inside a logarithm into addition, subtraction and multiplication outside it.

For $a>0$, $a\neq1$, $x>0$ and $y>0$:

$$\log_a(xy)=\log_a x+\log_a y$$ $$\log_a\left(\frac{x}{y}\right)=\log_a x-\log_a y$$

These rules come directly from exponent laws. If $\log_a x=r$ and $\log_a y=s$, then $x=a^r$ and $y=a^s$, so $xy=a^{r+s}$. The logarithm of $xy$ is therefore $r+s$, which is $\log_a x+\log_a y$.

Use the laws for products and quotients only, not for sums or differences. There is no law saying $\log_a(x+y)=\log_a x+\log_a y$. That made-up rule can wreck an otherwise correct solution very quickly.

Typical simplifications include:

$$\log_2 8+\log_2 4=\log_2 32=5$$

and

$$3\log_5 x-\log_5 y=\log_5\left(\frac{x^3}{y}\right)$$

provided $x>0$ and $y>0$.

Change of base

The change of base formula rewrites a logarithm in one base using logarithms in another base:

This helps because calculators usually give $\log$ and $\ln$ directly:

$$\log_4 7=\frac{\ln7}{\ln4}$$

It also helps in algebraic equations, where choosing one common base can turn a messy logarithmic expression into something that looks more like a polynomial-style equation. If you introduce a new variable, define it clearly, for example: let $u=\log_3 x$.

Solving exponential equations using logarithms

An exponential equation is an equation where the unknown appears in an exponent. There are two main approaches.

First, try writing both sides with the same base. For example:

$$\left(\frac{1}{3}\right)^x=9^{x+1}$$

can be rewritten as

$$3^{-x}=3^{2x+2}$$

so $-x=2x+2$ and $x=-\frac{2}{3}$.

Second, if a common base is awkward, take logarithms of both sides. For example:

$$2^{x-1}=10$$

Taking natural logarithms gives

$$\ln(2^{x-1})=\ln10$$

so

$$(x-1)\ln2=\ln10$$

and hence

$$x=1+\frac{\ln10}{\ln2}$$

The power law for logarithms does the key step here: it brings the unknown down from the exponent into ordinary multiplication.

For inequalities, remember that dividing by a negative number reverses the inequality sign. This matters when a logarithm such as $\log(0.9)$ appears, because $0<0.9<1$ means $\log(0.9)<0$. On a graph, the same idea connects to the behaviour of logarithmic and exponential functions, which is why the syllabus links this work to logarithmic and exponential graphs.

These methods are not just algebra games. They sit behind calculations involving pH and buffers in chemistry, and they help when estimating activation energy from experimental data where an exponential relationship has been transformed using logarithms.