# Scientific Notation and Logarithmic Scales

> Very large and very small numbers, significant figures, and the log scales behind decibels, earthquakes and pH.

From HSC Maths Advanced on WhizzNotes, free study notes for NSW students: https://whizznotes.com/mathematics-advanced/scientific-notation-and-log-scales/. Updated 3 November 2026.
Year 11 and Year 12 boxes go deeper than the Year 10 text around them.

### Story time: From atoms to galaxies
A human hair is about 0.000 08 m wide. The Sun is about 150 000 000 000 m from Earth. Writing all those zeros is slow, and it is easy to miscount them.

Scientists write these as $8 \times 10^{-5}$ m and $1.5 \times 10^{11}$ m. When numbers range from tiny to enormous, like the loudness of sounds or the strength of earthquakes, they go one step further and use a **logarithmic scale**, where each step means "ten times bigger".

**In short:** Scientific notation writes a number as $a \times 10^n$ with $1 \leq a < 10$, which makes very large and very small numbers easy to handle. On a logarithmic scale, like the Richter scale, decibels or pH, each step up means 10 times bigger.

## Starting point: Years 9 and 10

### Scientific notation

A number in **scientific notation** is written as

$$a \times 10^n \quad \text{where } 1 \leq a < 10 \text{ and } n \text{ is a whole number}$$

- Large numbers have a positive power: $45\,000 = 4.5 \times 10^4$.
- Small numbers have a negative power: $0.0032 = 3.2 \times 10^{-3}$.

Count how many places the decimal point moves. Moving left makes the power go up, moving right makes it go down.

### Worked example: Calculating
$$(3 \times 10^5) \times (4 \times 10^{-2}) = 12 \times 10^3 = 1.2 \times 10^4$$

$$\frac{8 \times 10^9}{2 \times 10^3} = 4 \times 10^6$$

Multiply the numbers, add the powers. Divide the numbers, subtract the powers. Then fix the front number so it is between 1 and 10.

### Calculator tip: The ×10ˣ key and rounding
Enter $3.2 \times 10^{-5}$ with the `×10ˣ` (or `EXP`) key: `3.2 ×10ˣ -5`. Avoid typing `× 10 ^ -5`, which can go wrong inside a division.

To round to significant figures, choose `Sci` in the setup menu and the number of figures. Choose `Norm` to go back to normal display.

### Significant figures

**Significant figures** are the digits that carry meaning. Count from the first non-zero digit.

- 3.047 has 4 significant figures.
- 0.0052 has 2 (the leading zeros only hold the place).
- $6.10 \times 10^5$ has 3 (the final zero was measured).

To round 186 420 to 2 significant figures: 190 000, or $1.9 \times 10^5$.

### Logarithmic scales

On a normal scale, equal steps **add** the same amount. On a **logarithmic scale**, equal steps **multiply** by the same amount, usually 10.

$$\log_{10} 10 = 1 \qquad \log_{10} 100 = 2 \qquad \log_{10} 1000 = 3$$

The log tells you the power of 10, so a log scale is really a scale of powers.

### Key idea: Three famous log scales
- **Earthquakes (Richter):** each step up means about 10 times more ground shaking.
- **Sound (decibels):** every 10 dB up means 10 times the sound intensity.
- **Acidity (pH):** $\text{pH} = -\log_{10}[\text{H}^+]$. Each step down means 10 times more acidic.

### Worked example: Comparing earthquakes
A magnitude 7 earthquake compared with a magnitude 5 is $10^{7 - 5} = 100$ times stronger in shaking.

### Watch out: Not double, but ten times
Going from magnitude 4 to 8 is not "twice as strong". It is $10^4 = 10\,000$ times.

**Mind map**

```text
Big and small numbers
  Scientific notation
    a × 10ⁿ, 1 ≤ a < 10
    Positive n: large
    Negative n: small
  Calculations
    Multiply: add powers
    Divide: subtract powers
  Significant figures
    Start at first non-zero
  Log scales
    Each step × 10
    Richter, decibels, pH
    Difference in log = power of 10
```

### Year 11: Working with log scale formulas
The **decibel** level of a sound with intensity $I$ is

$$L = 10\log_{10}\left(\frac{I}{I_0}\right)$$

where $I_0$ is the quietest sound people can hear. A sound 1000 times $I_0$ has $L = 10\log_{10}1000 = 30$ dB.

The log laws make comparisons easy. If sound B is 50 times as intense as A, then $L_B - L_A = 10\log_{10}50 \approx 17$ dB.

**pH.** Lemon juice has $[\text{H}^+] = 10^{-2.3}$, so its pH is 2.3. Pure water has pH 7, so lemon juice is $10^{4.7} \approx 50\,000$ times more acidic.

**Going backwards.** An earthquake 4 times as strong is $\log_{10} 4 \approx 0.6$ higher on the Richter scale. From a pH back to the concentration: $[\text{H}^+] = 10^{-\text{pH}}$, so pH 3.5 means $10^{-3.5} \approx 3.2 \times 10^{-4}$.

**Doubling.** If the intensity doubles, $L$ rises by $10\log_{10} 2 \approx 3$ dB, whatever the starting level.

**Other log formulas.** Read any formula the same way. In $R = 20\log_{10}\dfrac{N}{N_0}$, $R = 6$ means $\log_{10}\dfrac{N}{N_0} = 0.3$, so $N = 10^{0.3}N_0 \approx 2N_0$. In a base 2 formula such as $E = \log_2\dfrac{a}{t}$, halving $t$ raises $E$ by exactly 1.

### Year 12: Turning curves into straight lines
If $y = ab^x$, take logs of both sides:

$$\log_{10} y = \log_{10} a + x\log_{10} b$$

So plotting $\log_{10} y$ against $x$ gives a **straight line** with gradient $\log_{10} b$ and intercept $\log_{10} a$. Scientists use this to test whether data really is exponential: if the log plot is straight, the growth is exponential, and the gradient gives the growth factor.

**Using the line.** If the graph of $\log_{10} y$ against $x$ passes through $(0, 2)$ and $(5, 4)$, then $\log_{10} y = 2 + 0.4x$. At $x = 10$, $\log_{10} y = 6$, so $y = 10^6$.

**Log against log.** If $\log y$ against $\log x$ is a straight line with gradient $m$ and intercept $c$, then $\log y = m\log x + c$, which means $y = Ax^m$ with $\log A = c$. For base 5, gradient $-2$ and intercept 1 give $y = 5x^{-2} = \dfrac{5}{x^2}$.

## Quick summary

- Scientific notation: $a \times 10^n$ with $1 \leq a < 10$.
- Multiply: multiply the numbers and add the powers. Divide: divide and subtract the powers.
- Significant figures start at the first non-zero digit.
- On a log scale each step multiplies by 10. A difference of $d$ steps means $10^d$ times.
- Richter, decibels and pH are log scales.

## Quiz

Try these questions from the chapter quiz.

1. Write 72 000 in scientific notation.
   - A. $72 \times 10^3$
   - B. $7.2 \times 10^4$
   - C. $7.2 \times 10^3$
   - D. $0.72 \times 10^5$
   - Answer: B. $7.2 \times 10^4$. The front number must be between 1 and 10.

2. Write 0.000 45 in scientific notation.
   - A. $4.5 \times 10^4$
   - B. $4.5 \times 10^{-4}$
   - C. $45 \times 10^{-5}$
   - D. $4.5 \times 10^{-5}$
   - Answer: B. $4.5 \times 10^{-4}$. Move the point 4 places right.

3. Write $3.6 \times 10^{-2}$ as a decimal.
   - A. 360
   - B. 0.36
   - C. 0.036
   - D. 0.0036
   - Answer: C. 0.036. Move the point 2 places left.

The whole chapter quiz is on the page, for signed-in students: https://whizznotes.com/mathematics-advanced/scientific-notation-and-log-scales/#quiz