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Chapter 26: Problem 781

Use linear interpolation to find log \(5.723 .\)

Short Answer

Expert verified
Using linear interpolation, we find that log(5.723) lies between log(5.7) and log(5.8) with the values 0.755873 and 0.763428 respectively. Applying the linear interpolation formula, we calculate that log(5.723) 鈮� 0.757609.

Step by step solution

01

Find x1 and x2

Let's suppose that the logarithm of 5.723 lies between the values log(x1) and log(x2), such that x1 < 5.723 < x2. We can check the logarithmic values from the logarithm table, or use the calculator to estimate the closest logarithm values. From the log table, we find that the closest values are: - log(5.7) = 0.755873 - log(5.8) = 0.763428 So, x1 = 5.7 and x2 = 5.8.
02

Use Linear Interpolation

Now we'll use linear interpolation to find log(5.723). The linear interpolation formula is: \[f(x) = f(x_{1}) + \frac{f(x_{2}) - f(x_{1})}{x_{2} - x_{1}} (x - x_{1})\] In our case, the function f(x) represents the logarithm. So we need to find the value of f(5.723). We already know x1 = 5.7, x2 = 5.8, f(x1) = 0.755873 and f(x2) = 0.763428. So the formula becomes: \[log(5.723) = log(5.7) + \frac{log(5.8) - log(5.7)}{5.8 - 5.7} (5.723 - 5.7)\]
03

Calculate the Logarithm

Now we can calculate log(5.723): \[log(5.723) = 0.755873 + \frac{0.763428 - 0.755873}{5.8 - 5.7} (5.723 - 5.7)\] \[log(5.723) = 0.755873 + \frac{0.007555}{0.1} (0.023)\] \[log(5.723) = 0.755873 + 0.001736\] \[log(5.723) = 0.757609\] So, using linear interpolation, we find that log(5.723) 鈮� 0.757609.

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