82–89. Comparing growth rates Determine which of the two funct...

Question

82–89. Comparing growth rates Determine which of the two functions grows faster, or state that they have comparable growth rates.

ln x and log₁₀ x

Accepted Answer

Welcome back, everyone. In comparing the growth rates of the functions Y equals 5 LN 3 X and Y equals 5 log based 7 of 3 X for large values of X, which function grows faster or are their growth rates comparable? A says they have comparable growth rates. B says that 5 log-based 7 or 3 X grows faster than 5 LN3X because logarithms with smaller bases grow faster. C says 5 log based 7 of 3 X grows faster than 5 LN3X because logarithms with larger bases grow faster and B says the growth rates of both functions are not comparable because one uses a natural logarithm and the other uses a logarithm with base 7. Now how can we tell which grows faster or if their growth rates are comparable? Well, recall, OK. That If we have two functions, F and G. They have a comparable growth rate, OK? They have a comparable growth rate. If If the limit, as X approaches infinity of F X divided by G of X, the quotient is equal to M, where M is a positive and finite value. In other words, where M is between 0 and infinity. Now, if we let our two functions here, in other words, if we let uh F of X or function F relate to 5 and and 3 X. And if we let G of X be equal to 5 log base 7 of 3 X, OK. Then we can use this idea to see if we get a positive and finite value, and if we do, that means. Y equals 5 and then 3 X and Y equals 5 log based 7 of 3 X will have a comparable rate. So let's go ahead and compare them. With that in mind, Then we can find a limit as X approaches infinity of 5 LN 3 X. Divided by 5 log base 7 of 3 X. Now here, first notice that 5 Kansas 5. Next, if we think about it, it would be easier for us to compare these if they were in the same base for logarithms. So we can try to rewrite logbase 7 in terms of the natural log. When we do that, OK. Then, well, we still have the natural log of 3 X, but in our denominator, log based 7 of 3 X can be rewritten as the natural log of 3 X divided by the natural log of 7. No, we know we can rewrite that as the limit as x approaches infinity of the natural log of 3 X multiplied by the natural log of 7 divided by the natural log of 3 X. Where both of these will cancel. OK, and thus that will leave us finding the limit as x approaches infinity of the natural log of 7, which would be equal to the natural log of 7. This is a positive and finite value. Therefore, this tells us that the growth rates are comparable. If we go back to our answer choices, that means A is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

82–89. Comparing growth rates Determine which of the two functions grows faster, or state that they have comparable growth rates.

ln x and log₁₀ x

Key Concepts

Growth Rates of Functions

Natural Logarithm vs. Common Logarithm

Limits and L'Hôpital's Rule

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