Modeling Data with Exponential and Logarithmic Functions

By the end of this lesson, you will be able to determine whether data is better modeled by exponential or logarithmic growth, linearize data using logarithms, and fit exponential models to data points.

The Big Idea

Real-world data rarely comes pre-packaged in a formula. The skill is recognizing which function type fits the data pattern, using logarithms to linearize the data (convert it to a straight line), and then fitting a linear model — which is much simpler. This is the foundation of regression analysis in statistics.

Identifying the Model Type

Exponential growth/decay: ratios of consecutive terms are approximately constant.

$y = ab^x$: check if $y_{n+1}/y_n \approx b$ for all consecutive pairs.

Logarithmic growth: differences of $y$ values grow for equal ratios of $x$ values.

$y = a\ln x + b$: check if $\Delta y \approx a \cdot \Delta(\ln x)$.

Power function: $y = ax^b$ — ratios of $\log y$ values equal $b$ times ratios of $\log x$ values.

Linearizing Exponential Data

$y = ab^x$: take $\log$ of both sides:

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

Let $Y = \log y$, $A = \log a$, $B = \log b$:

$$Y = A + Bx$$

This is linear! Plot $\log y$ vs. $x$ — if you get a straight line, the data is exponential.

Recovering $a$ and $b$: $a = 10^A$ and $b = 10^B$.

Linearizing Power Data

$y = ax^b$: take $\log$ of both sides:

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

Let $Y = \log y$ and $X = \log x$:

$$Y = \log a + bX$$

Linear in $\log$-$\log$ space. Slope = $b$, intercept = $\log a$.

Example 1 — Exponential Fit from Two Points

Data: $(0, 5)$ and $(4, 80)$. Fit $y = ab^x$.

At $x = 0$: $a = 5$

At $x = 4$: $80 = 5b^4 \implies b^4 = 16 \implies b = 2$

Model: $y = 5 \cdot 2^x$

Example 2 — Linearization and Slope

Data points with roughly constant $\log y / \log x$ ratio:

$x$	$y$	$\log x$	$\log y$