Given that log 5.0 = 0.6990 and log 5.1 = 0.7076, find to the nearest hundredth a value of x for which log x = 0.7060. Can anyone solve this and provide an explanation? Thanks!
1 Answer
5.08
Explanation:
There's a few ways to do this, but either way, we will use the first order Taylor/Maclaurin expansion for
Let's define (with
We know two cases:
We will assume this is first order. We can use two log rules in order to simplify for
We could use our calculator to get a value for
Solving for
Plugging in the numbers given,
Therefore, we know the answer to 2 significant digits is 5.08.
With the linearity condition, there's actually a bit of a quicker way to reach it. Linear functions have a nice condition: if an input is
This agrees exactly with the above.
If we had just thrown the original into a calculator, we would have found that
showing the power of this method and that the error only appears two digits down, which is that second order term we originally mentioned.