Question #153489

Certain corresponding values of x and log10x are given below:

x: 300 304 305 307

log10 x: 2.4771 2.4829 2.4843 2.4871

Find log10 (310) by Lagrange’s formula.


1
Expert's answer
2021-01-11T18:03:35-0500

Here the intervals are unequal.


x0=300y0=2.4771x1=304y1=2.4829x2=305y2=2.4843x3=307y3=2.4871\begin{matrix} x_0=300 & y_0=2.4771 \\ \\ x_1=304 & y_1=2.4829 \\ \\ x_2=305 & y_2=2.4843 \\ \\ x_3=307 & y_3=2.4871 \\ \end{matrix}


By Lagrange’s interpolation formula we have

y=f(x)=(xx1)(xx2)(xx3)(x0x1)(x0x2)(x0x3)×y0y=f(x)=\dfrac{(x-x_1)(x-x_2)(x-x_3)}{(x_0-x_1)(x_0-x_2)(x_0-x_3)}\times y_0

+(xx0)(xx2)(xx3)(x1x0)(x1x2)(x1x3)×y1+\dfrac{(x-x_0)(x-x_2)(x-x_3)}{(x_1-x_0)(x_1-x_2)(x_1-x_3)}\times y_1

+(xx0)(xx1)(xx3)(x2x0)(x2x1)(x2x3)×y2+\dfrac{(x-x_0)(x-x_1)(x-x_3)}{(x_2-x_0)(x_2-x_1)(x_2-x_3)}\times y_2

+(xx0)(xx1)(xx2)(x3x0)(x3x1)(x3x2)×y3+\dfrac{(x-x_0)(x-x_1)(x-x_2)}{(x_3-x_0)(x_3-x_1)(x_3-x_2)}\times y_3



Put x=310x=310

f(310)=(310304)(310305)(310307)(300304)(300305)(300307)×2.4771f(310)=\dfrac{(310-304)(310-305)(310-307)}{(300-304)(300-305)(300-307)}\times 2.4771

+(310300)(310305)(310307)(304300)(304305)(304307)×2.4829+\dfrac{(310-300)(310-305)(310-307)}{(304-300)(304-305)(304-307)}\times 2.4829




+(310300)(310304)(310307)(305300)(305304)(305307)×2.4843+\dfrac{(310-300)(310-304)(310-307)}{(305-300)(305-304)(305-307)}\times 2.4843

+(310300)(310304)(310305)(307300)(307304)(307305)×2.4871+\dfrac{(310-300)(310-304)(310-305)}{(307-300)(307-304)(307-305)}\times 2.4871

=2.4914=2.4914

log10(310)=2.4914\log_{10}(310)=2.4914



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