Transforming Graphs using Logs

The transformation relies on the Product Rule of logarithms: $\log(xy) = \log x + \log y$. This allows the separation of a constant coefficient from the variable term.

The Power Rule of logarithms, $\log(x^n) = n \log x$, is essential for moving the variable from the exponent (in exponential models) or the power (in power models) into a linear multiplier position.

These laws apply regardless of the base used (common log $\log_{10}$ or natural log $\ln$), provided the base is consistent throughout the calculation.

For an exponential model of the form $y = Ab^{kx}$, taking the natural logarithm of both sides yields $\ln y = \ln(A) + kx \ln(b)$.

This matches the linear equation $Y = mX + c$ where the vertical axis $Y$ represents $\ln y$ and the horizontal axis $X$ represents $x$.

The gradient ($m$) of the resulting line is $k \ln b$, and the vertical intercept ($c$) is $\ln A$. To find the original constant $A$, one must calculate $e^c$.

For a power model of the form $y = ax^n$, taking the logarithm of both sides yields $\log y = \log a + n \log x$.

In this case, the linear form $Y = mX + c$ is achieved by plotting $\log y$ on the vertical axis against $\log x$ on the horizontal axis.

The gradient ($m$) of the line directly represents the power $n$, while the vertical intercept ($c$) represents $\log a$. The constant $a$ is recovered using the inverse log: $10^c$.

The critical difference lies in the horizontal axis: exponential models keep the independent variable $x$ in its original form, whereas power models require the logarithm of $x$.

If a plot of $\log y$ against $x$ is a straight line, the relationship is exponential. If a plot of $\log y$ against $\log x$ is a straight line, the relationship is a power law.

Identify the Axes First: Before performing any calculations, check the labels on the axes. If only the y-axis is logarithmic, it is an exponential model. If both axes are logarithmic, it is a power model.

Base Consistency: Always use the same base for the gradient and the intercept. If the graph uses $\ln y$, use $e$ to find the constants; if it uses $\log_{10} y$, use $10$.

Intercept Interpretation: The intercept $c$ is the value of the log-variable, not the constant itself. You must always perform the inverse operation (e.g., $A = 10^c$) to find the physical constant.

Units and Scale: Ensure that any units provided in the problem are handled before taking logarithms, as you cannot take the log of a value with units (logs are dimensionless).

Mixing up $x$ and $\log x$: Students often plot $\log y$ against $x$ for a power model, which results in a curve rather than the intended straight line.

Negative Values: Logarithms are only defined for positive numbers. If the data contains zero or negative values, a simple log transformation cannot be applied without first shifting the data.

Gradient Miscalculation: For exponential models $y = Ae^{kx}$, the gradient is exactly $k$. For $y = Ab^x$, the gradient is $\log b$. Forgetting which form is being used leads to incorrect constant values.