---
title: "Fit y = a b^x (growth data)"
subject: "Numerical Method"
chapter: 4
key: "semester_4/numerical_method:fit-y-abx-d2"
source: https://www.bctengineer.com/q/semester_4/numerical_method/fit-y-abx-d2
site: BCT Engineering
---

# Fit y = a b^x (growth data)

Numerical Method · Chapter 4: Interpolation

- Asked in: 2068 Bhadra, 8 marks
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/numerical_method/fit-y-abx-d2
- Whole chapter: https://www.bctengineer.com/md/semester_4/numerical_method/chapter-4

## Question

Find a curve of the form y = ab^x that fits the following set of observations using least square method.
| x | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| y | 1.2 | 2.5 | 6.25 | 15.75 | 28.65 |

## Short answer

Linearize to log y = log a + x log b. Solving the normal equations yields log a = -0.2811, log b = 0.3555, so a = 0.5235, b = 2.2674. The fitted curve is y = 0.5235(2.2674)^x.

## Answer

Take logarithms (base 10): y = a·bˣ → log y = log a + x·log b. Put Y = log y, A = log a, B = log b, so Y = A + Bx is a straight line in x.

| x | y | Y = log y | x² | xY |
|---|---|---|---|---|
| 1 | 1.2 | 0.0792 | 1.0000 | 0.0792 |
| 2 | 2.5 | 0.3979 | 4.0000 | 0.7959 |
| 3 | 6.25 | 0.7959 | 9.0000 | 2.3876 |
| 4 | 15.75 | 1.1973 | 16.0000 | 4.7891 |
| 5 | 28.65 | 1.4571 | 25.0000 | 7.2856 |
| Σx = 15 |  | 3.9274 | 55.0000 | 15.3374 |

#### Normal equations

```text
ΣY = n·A + B·Σx      →  3.9274 = 5A + 15.0000B
ΣxY = A·Σx + B·Σx²   →  15.3374 = 15.0000A + 55.0000B
```

Solving: B = (nΣxY − ΣxΣY)/(nΣx² − (Σx)²) = 0.3555,  A = (ΣY − BΣx)/n = -0.2811.

a = 10ᴬ = 0.5235, b = 10ᴮ = 2.2674.

**Answer: y = 0.5235 · (2.2674)ˣ**  (growth curve, b > 1; at x = 5 it gives 31.37 against 28.65).

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. How is y = a b^x fitted?
   A. Fit a parabola
   B. Use Lagrange
   C. Fit a line to log y against x
   D. Fit a line to y against x
   Answer: C. Y = A + Bx. (Key point: Method)

2. B = log b = 0.3555. What is b?
   A. 0.3555
   B. 10^0.3555 = 2.2674
   C. 1/0.3555
   D. ln 0.3555
   Answer: B. b = 10^B. (Key point: B)

3. What does b > 1 indicate?
   A. Oscillation
   B. Constant
   C. Decay
   D. Growth
   Answer: D. y increases with x. (Key point: Growth)

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