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Problem 43 Hard Difficulty

Use Formula 10 to graph the given functions on a common screen. How are these graphs related?

$ y = \log_{1.5} x $ , $ y = \ln x $ , $ y = \log_{10} x $ , $ y = \log_{50} x $

Answer

To graph these functions, we use $\log _{1.5} x=\frac{\ln x}{\ln 1.5}$ and $\log _{30} x=\frac{\ln x}{\ln 50}$.
These graphs all approach $-\infty$ as $x \rightarrow 0^{+}$, and they all pass through the
point $(1,0) .$ Also, they are all increasing, and all approach $\infty$ as $x \rightarrow \infty$
The functions with larger bases increase extremely slowly, and the ones with
smaller bases do so somewhat more quickly, The functions with large bases
approach the $y$ -axis more closely as $x \rightarrow 0^{+}$.

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Video Transcript

all right for this problem, we're going to be using a graphing calculator. And just in case your calculator isn't able to type in something like Log based 1.5 of X or log based 50 of X, you can use the change of based formula. And so, for log based 1.5 of X, you can change it to natural log of X over natural log of 1.5. It would also work to do common log of X over common laudable in 0.5, and similarly, for log based 50 of X, you could do natural log of X over natural log of 50. Now, newer calculators are able to do other bases, like Log based 1.5 or log based 50. But just in case you don't have a calculator like that, we should all be able to do it this way. So we grab a calculator, the Goethe y equals, and we type in all of these functions the four functions that were comparing. And then we want to view them on a common screen so you can fiddle around with your window dimensions until you get something that you're happy with. I decided to use negative 1 to 10 on my X axis, and I didn't change my Y values. So just negative 10 to 10. And here's how it looks. Okay. And if you press trace, you can go through and see which color is witchcraft. So the blue one was log based 1.5, and the red one is log based E natural logs, and the black one was logged based 10 and the pink one was logged based 50. So we can see that as the base girls bigger and bigger, the graph is more and more flat.