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Problem 45 Hard Difficulty

Suppose that the graph of $ y = \log_2 x $ is drawn on a coordinate grid where the unit of measurement is an inch. How many miles to the right of the origin do we have to move before the height of the curve reaches 3 ft?

Answer

1084587.7 $\mathrm{mi}$

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Video Transcript

for this problem. We have the graph of y equals log based two of X, and I've made a rough sketch of it. We know it goes through the X axis at the 20.10 and we're interested in finding out what its X corn it is. When the Y coordinate is three feet, every unit is one inch, so three feet will be 36 inches. So what is the X coordinate when the Y coordinate is 36 inches? How far out would we have to go? Well, what we can do is take our function log based two of X and set it equal to 36. Substitute 36 in for why and when we saw that we get X equals two to the 36th Power. Now, if you put that in your calculator, you get a very, very large number. It's it changes it to scientific notation. I can only write part of its 6.87 times 10 to the 10th inches. Now let's convert that into miles, so it'll be easier to manage. So to convert it into feet, we would divide it by 12. Now we have feet and to convert it to Miles. We would divide that by 5280 because there are 5280 feet in a mile, and once we divide by both of those numbers, we end up with one million 84,000 587 0.7 miles. That's how far you would have to go to reach a height of three feet on this graph. That is pretty extreme. That shows us just how slowly this graph is growing.

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