How is y(7) obtained?

Substitute x = 7 in y = 23.2761 (0.8428)^x

Fit y = a b^x and find y(7)

Short answer

Take log: Y=log y. Normal equations yield A=1.3669, B=−0.0743, so a=23.2761, b=0.8428. Curve: y=23.2761·0.8428ˣ. y(7)≈7.031.

Fit the following set of data to a curve of the form y = a b^x. Also evaluate y(7).

x	2	4	6	8	10	12
y	16.0	11.1	8.7	6.4	4.7	2.6

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
2	16	1.2041	4.0000	2.4082
4	11.1	1.0453	16.0000	4.1813
6	8.7	0.9395	36.0000	5.6371
8	6.4	0.8062	64.0000	6.4494
10	4.7	0.6721	100.0000	6.7210
12	2.6	0.4150	144.0000	4.9797
Σx = 42		5.0822	364.0000	30.3767

ΣY = n·A + B·Σx      →  5.0822 = 6A + 42.0000B
ΣxY = A·Σx + B·Σx²   →  30.3767 = 42.0000A + 364.0000B

Solving: B = (nΣxY − ΣxΣY)/(nΣx² − (Σx)²) = -0.0743, A = (ΣY − BΣx)/n = 1.3669.

a = 10ᴬ = 23.2761, b = 10ᴮ = 0.8428.

Answer: y = 23.2761 · (0.8428)ˣ

y(7) = 23.2761 × (0.8428)⁷ = 7.0313

y(7) ≈ 7.031

✓ Answer checked by Claude Opus 5.5 (high effort)

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