@whpalmer4
Use log$$\log_{2} 3 \approx 1.5850 and \log_{2} 5\approx 2.3219 \to approximate the value of each expression.$$

Question

@whpalmer4 
Use log$$\log_{2} 3 \approx 1.5850 and \log_{2} 5\approx 2.3219 \to approximate the value of each expression.$$

anonymous · Answer

1) $$\log_{2}  25$$

whpalmer4 · Answer

how could you write 25 using 5 and 3 and multiplication and division?

anonymous · Answer

well is actually suppose to represent two squiggly lines sort of like an equal sign

anonymous · Answer

http://www.clker.com/cliparts/2/9/a/0/1349815143552846380Wavy%20Lines.svg.med.png

whpalmer4 · Answer

Yeah, $\approx$ means approximately

But the key to this problem is figuring out how to express 25 using only 5 and 3 and multiplication and division (may not have to use all of them)

whpalmer4 · Answer

Hint: what is 5*5?

anonymous · Answer

25

preetha · Answer

Whpalmer - good work there!

whpalmer4 · Answer

Right.  Remember the property of logarithms involving multiplication?  
$$\log (a*b) = \log a + \log b$$

whpalmer4 · Answer

@Preetha thanks, boss ;-)

anonymous · Answer

but that's not the answer right

whpalmer4 · Answer

We can write $$\log 25 = \log (5*5) = \log 5 + \log 5$$right?  Works for any base...

anonymous · Answer

but only for this particular problem

whpalmer4 · Answer

No, it's always true.  For logarithms of any base $b$, $$\log_b (x*x) = \log_b x +\log_b x = 2\log_b x$$

anonymous · Answer

no no I understand that rule but as for the 25 and 5

whpalmer4 · Answer

we happen to know the value of $\log_2 5$, so we can use that to our advantage:
$$\log_2 25 = \log_2 (5*5) = \log_2 5 + \log_2 5 = 2*\log_2 5 \approx 2*2.2319 \approx 4.6439$$

whpalmer4 · Answer

We could also write it like this:
$$\log_2 25 = \log_2 5^2 = 2\log_2 5$$because $$\log_b x^a = a\log_b x$$

anonymous · Answer

ooohhhh ok I get it now

whpalmer4 · Answer

really, the whole trick here was realizing how to "construct" 25 from the building blocks we have, 5 and 3

whpalmer4 · Answer

$$\log_2 75 = \log_2 (3*25) = \log_2 3 + 2\log_2 5 = 1.5850 + 4.6439 = 6.22882$$
(for example)

anonymous · Answer

and since 5 is a factor of 25 we choose that as the number they gave us

anonymous · Answer

I got cha :)

whpalmer4 · Answer

as I mentioned earlier, I memorized a handful of log values, and I can use them like tinker toys to estimate logs of other numbers, just like I did with 25 and 75 here.

there's another property of logs which is useful, called the base change property:

$$\log_b x = \frac{\log_a x}{\log_a b}$$

So if you know the log in one base, you can find it in another base by some simple arithmetic.

whpalmer4 · Answer

For example, if you wanted to compute the log base 2 of 25, but didn't have those values handy, but did have a calculator that does $\log_{10}$, you could do this:

$$\log_2 25 = \frac{\log_{10} 25}{\log_{10}2} = \frac{1.39794}{0.30103}  = $$(go ahead, do it!)

anonymous · Answer

i think she was telling us about that

whpalmer4 · Answer

so, what do you get when you do that division on your calculator?

anonymous · Answer

4.6439

whpalmer4 · Answer

Right.  Does that answer look familiar? :-)

anonymous · Answer

yep lol we just took a short cut :)

whpalmer4 · Answer

The fun thing about problems with logarithms (to the extent that there is a fun thing) is that often there are many different ways to get to the answer.  It can also be a frustrating thing, if you can't figure out how to do any of them!

anonymous · Answer

exactly which was how I was before you taught me

whpalmer4 · Answer

Sounds like a satisfied customer :-)

anonymous · Answer

definitely

whpalmer4 · Answer

was that all of these problems, or have you done the rest of them while I've been faffing about here trying to juggle multiple questions?

anonymous · Answer

Oh trust me I have plenty more to do she gave us like 3 worksheet but I don't want to bother you anymore because I don't want you to get tired and other ppl may need help :)

anonymous · Answer

@whpalmer4

whpalmer4 · Answer

Well, I keep hopping off to help others as they summon me (just like you do).  It's more likely at this hour of the day (7PM my time) that you'll get tired than that I will :-)

anonymous · Answer

lol so ur saying I can ask you more lol

anonymous · Answer

@whpalmer4

whpalmer4 · Answer

Yes.  By the way, I get a notification each time you post in the thread, no need to also tag me unless you grow impatient :-)

anonymous · Answer

oh my bad lol I'm sorry ok here we go 
solve each inequality or equation. Round to the nearest ten-thousandth
9) 3^x>243