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britz821x16 · Feb 14, 2012, 08:09 PM

2.find the equation of the line tangent to the graph of f(x)=(lnx)^4 at x=5?

jcaron2 · Feb 14, 2012, 08:36 PM

First you need to compute the derivative to find the slope at x=5.

$f'(x)=\frac{d}{dx}\[(\ln x)^4\] = 4(\ln x)^3\;\cdot\;\frac{d}{dx}(\ln x)=\frac{4(\ln x)^3}{x}$

$f'(5)=\frac{4(\ln 5)^3}{5}$

Once you have the slope, you just need to find a point on the line. The obvious choice is to use the point [5,f(5)]:

$f(5)=(\ln 5)^4$

Now just write the equation of a line in point-slope form:

$y-y_0=m(x-x_0)$

$y-(\ln 5)^4=\frac{4(\ln 5)^3}{5}(x-5)$

$y=\frac{4(\ln 5)^3}{5}x-4(\ln 5)^3+(\ln 5)^4$

britz821x16 · Feb 14, 2012, 09:15 PM

What is the answer to f(x)=(lnx)^5 at x=4?

jcaron2 · Feb 14, 2012, 09:36 PM

f(x)=(lnx)^5 at x=4 is (ln4)^5. It can't be simplified more than that.

britz821x16 · Feb 14, 2012, 09:46 PM

I know question 2 is y=3.34x-9.97, but I don't know how to plug (ln4)^5 in a equation

jcaron2 · Feb 14, 2012, 10:18 PM

Well, thanks for switching my reputation back to "helpful". :)

Since you have the answer to question #2, I edited my original post to answer that one.

Now I'll show you how to evaluate (4(ln 5)^3)/5. If you use the calculator built into Windows (in Accessories), you can change it to a Scientific calculator (under the View menu), and then you can plug in the expression. (It should be similar on almost any other scientific calculator, whether stand-alone, on your computer, on your smartphone, or online at a site like this one).

Type in 5, then hit ln. That gives you ln(5) = 1.61

Then hit the $x^y$ button, followed by 3, followed by =.

That gives you (ln(5))^3, which is around 4.17.

Now hit * 4 / 5 = which gives you (4(ln 5)^3)/5 = 3.34, which agrees with the answer you have!