AHL 2.10—Scaling large numbers, log-log graphs Notes

Scaling Large Numbers with Logarithms

1. In mathematics and data analysis, dealing with very large or very small numbers can be challenging.
2. Logarithms provide a powerful tool for scaling such numbers to more manageable sizes.

Basic Concept

The logarithm of a number is the exponent to which a base (usually 10) must be raised to produce that number.

Mathematically, if

$$\log_a(x)=n \iff x=a^n$$

This property allows us to represent large numbers with smaller, more manageable values:

• 1,000 becomes 3
• 1,000,000 becomes 6
• 0.001 becomes -3

Key Properties

• $\log _a(1)=0$ since $a^0=1$.
• $\log _a(a)=1$ since $a^1=a$.
• $\log _a(x y)=\log _a(x)+\log _a(y)$ (logarithm product rule).
• $\log _a\left(x^b\right)=b \log _a(x)$ (logarithm power rule).
• $\log _a\left(\frac{x}{y}\right)=\log _a(x)-\log _a(y)$ (logarithm quotient rule).
• Change of Base Formula:
  $\log _a(x)=\frac{\log _b(x)}{\log _b(a)}$ for any valid base $b$.

Natural Logarithm

The natural logarithm (denoted as $\ln(x)$)) is a logarithm with base $e$, where $e \approx 2.718$ is the mathematical constant known as Euler’s number.

Definition

The natural logarithm is defined as:

$$\ln(x) = n \iff e^n = x$$

Key Properties

• $\ln (1)=0$ since $e^0=1$.
• $\ln (e)=1$ since $e^1=e$.
• $\ln (a b)=\ln (a)+\ln (b)$ (logarithm product rule).
• $\ln \left(a^b\right)=b \ln (a)$ (logarithm power rule).
• $\ln \left(\frac{a}{b}\right)=\ln (a)-\ln (b)$ (logarithm quotient rule).

Choosing Appropriate Scales

Wide Range of Values

When data contains a wide range of values, choosing an appropriate scale is crucial for meaningful visualization and analysis.

Linear Scale Limitations

• A linear scale may not effectively represent data with a wide range.
• For example, if plotting values from 1 to 1,000,000, smaller values might be indistinguishable.

Logarithmic Scale Advantages

• A logarithmic scale can effectively display data spanning several orders of magnitude.
• Each step on the axis represents a multiplication by a constant factor (usually 10).

Emphasizing Rate of Growth

When the focus is on the rate of growth rather than absolute values, logarithmic scales are particularly useful.

Linearizing Data with Logarithms

Exponential Relationships

For data following an exponential relationship ($y = ae^{bx}$), taking the natural logarithm of both sides yields:

$$\ln(y) = \ln(a) + bx$$

This transforms the exponential relationship into a linear one, where $\ln(y)$ is plotted against $x$.

Power Relationships

For data following a power relationship ($y = ax^b$), taking the logarithm (base 10 or natural) of both sides yields:

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

This transforms the power relationship into a linear one, where $\log(y)$ is plotted against $\log(x)$.

Using Best-Fit Straight Lines

After linearizing data, a best-fit straight line can be used to determine the parameters of the original relationship.

For Exponential Relationships

In the linearized form $\ln(y) = \ln(a) + bx$:

• The y-intercept gives $\ln(a)$
• The slope gives $b$

For Power Relationships

In the linearized form $\log(y) = \log(a) + b\log(x)$:

• The y-intercept gives $\log(a)$
• The slope gives $b$

Interpreting Log-Log Graphs

• Log-log graphs plot the logarithm of y against the logarithm of x.
• They are particularly useful for power law relationships.

Key Features

1. A straight line on a log-log plot indicates a power law relationship.
2. The slope of the line gives the exponent in the power law.
3. Parallel lines indicate relationships that differ by a constant factor.

Interpreting Semi-Log Graphs

• Semi-log graphs plot the logarithm of one variable against the linear scale of another.
• They are particularly useful for exponential relationships.

Key Features

1. A straight line on a semi-log plot (with log scale on y-axis) indicates an exponential relationship.
2. The slope of the line is related to the growth or decay rate.
3. Parallel lines indicate relationships that differ by a constant factor.