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IB Mathematics AI AHL Log Graphs MAI Study Notes - New Syllabus

IB Mathematics AI AHL Log Graphs MAI Study Notes

LEARNING OBJECTIVE

  • To use logarithms to scale data, linearize exponential and power relationships, and interpret log-log and semi-log graphs.

Key Concepts: 

  • Scaling using logarithms
  • Linearizing data
  • Interpreting log-log and semi-log graphs

MAI HL and SL Notes – All topics

Scaling Using Logarithms

When data spans a very large or very small range, plotting it on a linear scale can make trends difficult to see. A logarithmic scale solves this by plotting the logarithm of values instead of the raw values themselves.

Why use logarithmic scales?

  • Makes it easier to visualize exponential growth or decay

  • Compresses large numbers and spreads out smaller numbers

  • Useful for data involving powers of 10, such as:

    • Earthquake magnitudes (Richter scale)

    • Sound intensity (decibels)

    • pH levels

    • Population growth or radioactive decay

How it works:

In a log scale:

  • Equal distances on the axis represent equal ratios, not differences.

  • For example, on a base-10 log scale:

    • The distance between 1 and 10 is the same as between 10 and 100.

    • Each tick increases by a factor of 10.

Example

A piece of paper is 0.075 mm. We fold it repeatedly (beyond the usual limit of 7).

Use a graph to find when it becomes 1 cm thick, then how many folds until it reaches the moon?

▶️Answer/Explanation

Solution:

We graph $h(n) = 0.075(2^n)$; Moon $≈ 42$ folds

LINEARISATION OF THE LOGARITHMIC MODEL

Consider the logarithmic model:

\( y = a + b \ln x \).

If we set \( X = \ln x \), the model takes the form: \( y = a + bX \) [linear]. 

Example

Graph the height of the paper against no. of folds on a ln(h) against n graph:

nh
10.15
20.3
30.6
……
207864.2
42329.8 bn
▶️Answer/Explanation

Solution:

We need to take each h, and do ln(h)

nhln(h)
10.15-1.897
20.3-1.204
30.6-0.51
………
207864.211.27
42329.8 bn26.52

 

LINEARISATION OF THE EXPONENTIAL MODEL

Consider the exponential model: \( y = Ae^{kx} \).

Apply \( \ln \): \( \ln y = \ln A + kx \).

If we set \( Y = \ln y \), the model takes the form: \( Y = \ln A + kx \) [linear]. 

Example

For the following data, showing the mass of bacteria (M, grams) after t hours, show that it should fit an exponential model, giving the equation for M in terms of t:

t (hrs)0123456810
M (g)2.73.34.25.06.47.5
▶️Answer/Explanation

Solution:

Semi-log graph:

On GDC, we find a ln(M) column:

t (hrs)0.9931.4351.1941.6091.8562.0152.1862.6533.077

Very close to linear, so the original data is exponential.
 Using GDC `LinReg(mx + b)` function with x-list = t & y-list = ln(M).
 $r^2 = 0.999$, which confirms exponential correlation.

GDC also shows:

 $m = 0.2065$
 $b = 0.9964$

Thus:

$
\ln M = 0.2065t + 0.9964
$

$
M = e^{0.2065t + 0.9964} = e^{0.9964} \cdot e^{0.2065t}
$

$
M = 2.7084 \cdot e^{0.2065t} \approx 2.71 \cdot (1.23)^t
$

Conclusion:
Which shows it can be modelled with an exponential model. This one is growing by approximately $23\%$ per hour

LINEARISATION OF THE POWER FUNCTION MODEL
Consider the power function model: \( y = kx^n \).

Apply \( \ln \): \( \ln y = \ln k + n \ln x \).

If we set \( X = \ln x \) and \( Y = \ln y \), it takes the form: \( Y = \ln k + nX \) [linear]. 

Example

A group of students try to model the motion of a football being dropped off their HS roof terrace

Time (s)Distance Fallen (m)
0.51.3
15
1.511
219
2.529
343
3.562.5
▶️Answer/Explanation

Solution:

  Normal scatter plot:

$f$ vs. $t$

Not linear, therefore not exponential.

Try $\ln(f)$ vs. $t$ plot:

Not linear, therefore not exponential.

Try $\ln(f)$ vs. $\ln(t)$ plot:

Approximately linear, therefore suggests a power series model.

Linear regression on $\ln(f)$ vs. $\ln(t)$ using GDC:

We get:

$m = 1.966$
$b = 1.608$

$
\ln f = 1.966 \ln t + 1.608
$

$
f = e^{1.966 \ln t + 1.608} = e^{1.966 \ln t} \cdot e^{1.608}
$

$
f = e^{1.608} \cdot t^{1.966} \approx 4.99 \cdot t^{1.97}
$

This is a power model.

SEMI- LOG AND LOG GRAPHS

A semi-log graph is a type of graph that uses a logarithmic scale for one axis (usually the y-axis) and a linear scale for the other axis (usually the x-axis). A log-log graph, on the other hand, uses a logarithmic scale for both the x-axis and the y-axis.

Semi-log Graph

 

In a semi-log graph, the y-axis is scaled logarithmically, and the x-axis is scaled linearly. This means that each unit on the y-axis represents a tenfold increase in the value, while the x-axis increases by equal intervals.

Semi-log graphs are particularly useful when plotting data that spans several orders of magnitude or when the data follows an exponential trend. The most common example is exponential growth or decay, such as population growth, radioactive decay, or the spread of a virus.

For example, consider the exponential function:

$y = ab^x$

Taking the logarithm of both sides gives:

$\log(y) = \log(a) + x \log(b)$

This equation is linear in x, which means that when y is plotted on a logarithmic scale and x is plotted on a linear scale, the graph will be a straight line.

Log-log Graph

In a log-log graph, both the x-axis and the y-axis are scaled logarithmically. This type of graph is used when both variables cover a large range of values and when the relationship between the variables is a power law.

A power law has the form:

$y = ax^k$

Taking the logarithm of both sides gives:

$\log(y) = \log(a) + k \log(x)$

This is a linear equation in \(\log(x)\), meaning that if both axes are logarithmic, the graph will be a straight line. The slope of the line corresponds to the exponent \(k\), and the y-intercept corresponds to \(\log(a)\).

Log-log graphs are often used in physics, engineering, and other sciences to analyze relationships between variables that span many orders of magnitude.

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