IB Math AI SL/HL Paper 1 Exam Style Practice Questions: Logarithms

Sound Intensity and Logarithmic Functions

IB Mathematics HL • Logarithms • Exponential Functions • Modelling in Context

Lesson Overview

In this lesson, we use a logarithmic model to relate the loudness of a sound, measured in decibels (dB), to its intensity, measured in watts per square metre.

The model is

$L=10\log_{10}\left(\frac{I}{I_0}\right), \qquad I>0$

where

$I_0=10^{-12}\text{ Wm}^{-2}$

is the reference intensity.

This type of question is common in IB Mathematics because it combines logarithms, scientific notation, and interpretation of change in a real-world context.

⭐ Key Concepts

The sound intensity model is

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

To find loudness from intensity, substitute directly into the formula.
To find intensity from loudness, rearrange using powers of 10.
A change in decibels corresponds to a multiplicative change in intensity, not an additive one.

Useful logarithm facts:

$\log_{10}(a^n)=n\log_{10}(a)$

$\log_{10}(ab)=\log_{10}(a)+\log_{10}(b)$

$\log_{10}\left(\frac{a}{b}\right)=\log_{10}(a)-\log_{10}(b)$

📘 Clear IB-Style Explanation

The decibel scale is logarithmic, which means that equal increases in loudness do not correspond to equal increases in intensity.

For example, if the loudness increases by 10 dB, the intensity is multiplied by 10. If the loudness increases by 20 dB, the intensity is multiplied by 100.

In IB questions, students are usually expected to:

substitute values carefully into the logarithmic formula,
rearrange logarithmic equations correctly,
write answers in scientific notation when required,
interpret what a change in decibels means for intensity.

📌 Worked Example 1 — Finding Loudness from Intensity

A concert speaker produces sound with intensity

$I=3.2\times10^{-4}\text{ Wm}^{-2}$

Given that

$I_0=10^{-12}\text{ Wm}^{-2}$

Find the loudness of the sound, correct to 1 decimal place.

Solution

Use

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

Substitute the values:

$L=10\log_{10}\left(\frac{3.2\times10^{-4}}{10^{-12}}\right)$

$=10\log_{10}(3.2\times10^8)$

Now use logarithm laws:

$L=10\left(\log_{10}(3.2)+\log_{10}(10^8)\right)$

$=10\left(\log_{10}(3.2)+8\right)$

$\approx10(0.50515+8)$

$\approx10(8.50515)=85.0515$

So the loudness is

$85.1\text{ dB}$

📊 Marks: M1 A1

📌 Worked Example 2 — Finding Intensity from Loudness

Another sound has loudness

$142\text{ dB}$

Find its intensity. Give your answer in the form $a\times10^k$ , where $1\le a<10$ and $k\in\mathbb{Z}$ .

Solution

Start with

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

Substitute $L=142$ :

$142=10\log_{10}\left(\frac{I}{10^{-12}}\right)$

$14.2=\log_{10}\left(\frac{I}{10^{-12}}\right)$

Rewrite in exponential form:

$\frac{I}{10^{-12}}=10^{14.2}$

$I=10^{-12}\times10^{14.2}$

$I=10^{2.2}$

Now convert to scientific notation:

$10^{2.2}=10^2\times10^{0.2}$

$=100\times1.58489\ldots$

$\approx158.489$

$I\approx1.58\times10^2\text{ Wm}^{-2}$

📊 Marks: M1 M1 A1

📌 Worked Example 3 — Factor Change in Intensity

The loudness of a machine increases from

$96\text{ dB}$

$111\text{ dB}$

Find the factor by which the intensity increases.

Solution

Let the original intensity be $I_1$ and the new intensity be $I_2$ .

Using the model for both sounds:

$96=10\log_{10}\left(\frac{I_1}{I_0}\right)$

$111=10\log_{10}\left(\frac{I_2}{I_0}\right)$

Subtract the two equations:

$111-96=10\log_{10}\left(\frac{I_2}{I_1}\right)$

$15=10\log_{10}\left(\frac{I_2}{I_1}\right)$

$1.5=\log_{10}\left(\frac{I_2}{I_1}\right)$

Convert to exponential form:

$\frac{I_2}{I_1}=10^{1.5}$

$\frac{I_2}{I_1}\approx31.6$

So the intensity increases by a factor of

$31.6$

📊 Marks: M1 A1

⚠ Common Mistakes

Forgetting to divide by the reference intensity $I_0$ inside the logarithm.
Using natural logarithms incorrectly without adjusting the formula.
Treating a change in decibels as an additive change in intensity instead of a multiplicative change.
Giving the intensity answer in ordinary form when the question asks for scientific notation.
Rounding too early, especially in multi-step logarithm calculations.

📘 IB Exam Tips

Always write down the original formula before substituting.
When solving for intensity, divide by 10 first before converting from logarithmic to exponential form.
Use the log laws clearly if showing exact algebraic working.
For “factor increase” questions, compare the two equations rather than finding each intensity separately unless needed.
Check units carefully: loudness is in dB, intensity is in $\text{Wm}^{-2}$ .