If logarithms didn't exist, Satan would have invented them

guernica

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Given the following equation:
(S/V+t)*ln(S/V+t)+(S/V+t)*-ln(S/V)=S+Vt
with S and V as constants, my TI-89, whom I never dare contradict, says that
t=S*(e^(V) - 1)/V and S/V > 0.

Having dealt with logarithms only in much simpler settings (exponential growth and decay), I'm stymied as to how to proceed. I tried the obvious, like dividing by (S/V + t), but merely entered into new realms of complexity as a result. I would throw myself to the ground in groveling gratitude if someone could lay out each (baby) step of the journey to
t=S*(e^(V) - 1)/V, leaving as little to the imagination as possible.
 
After "dividing by (S/V + t)", were you left with: ln(S/V + t) - ln(S/V) = V :?:

"If logarithms didn't exist, Satan would have invented them" ; hmmm...perhaps V = 666 :shock:
 
To be certain..is that \(\displaystyle \frac{S}{V}+t\) or \(\displaystyle \frac{S}{V+t}\)?.
 
It's a matter of manipulating the exponent laws.

\(\displaystyle (\frac{S}{V}+t)ln(\frac{S}{V}+t)-(\frac{S}{V}+t)ln(\frac{S}{V})=S+Vt\)

Always look to factor out something common:

\(\displaystyle (\frac{S}{V}+t)\left[ln(\frac{S}{V}+t)-ln(\frac{S}{V})\right]=S+Vt\)

Divide by what is factored out:

\(\displaystyle ln(\frac{S}{V}+t)-ln(\frac{S}{V})=\frac{S+Vt}{\frac{S}{V}+t}\)

But the right side of the equality simplifies down to just V:

The left side, use exponent law:

\(\displaystyle ln(\frac{(\frac{S}{V}+t)V}{S})=V\)

\(\displaystyle \frac{(\frac{S}{V}+t)V}{S}=e^{V}\)

\(\displaystyle \frac{S}{V}+t=\frac{Se^{V}}{V}\)

\(\displaystyle t=\frac{Se^{V}}{V}-\frac{S}{V}\)

\(\displaystyle \boxed{t=\frac{S(e^{V}-1)}{V}}\)
 
galactus: Even the most demanding supplicant (like me) can only smile contentedly at your brisk march to the oasis, and can only cringe disgustedly at my own
little lapse in handling the logarithm that led to my desolate wanderings in the desert.

denis: I liked v=666. If compelled to traffic in logarithms in the future, I intend to have an exorcist on retainer.
 
Sure agree; anyhow, **** sure is much more enjoyable than logs; here's my proof:

HELLUVA HOT STUFF
=================
A truly versatile word. The things you can do with it are simply amazing.
For instance, you can:
1- beat it out of someone
2- couple it with high water
3- raise it
4- watch bats come out of it
...plus a whole repertoire of other ingenious permutations and combinations.

If you're asking yourself "what the **** is he talking about?", then you've
already used it; yes: ****'s the word. May I repeat: an amazingly versatile
4 letter word. A word that adds knocks and kicks to utterances. Think of how
weak these beauties would be without it:
1- get the **** out of here
2- I've had one **** of a day
3- what the **** is going on?

In the days of Nero, Pilate, angels and traffic stars, when Earth was the
floor of a gigantic sky-dome, back then **** was known as "Hades", and was
situated in Earth's cellar. When Galileo astronomated in on the scene and
changed Earth into a spinning globe rotating in space, it became necessary
to change Hades' name and location.

Theologians re-baptized it "****". However, its exact location is still
under review. Therefore, we're currently not sure where Hitler is, or where
Jimmy Swaggart, Madonna and Guernica are going!

Let's take a moment of silence to be grateful for this name change. Can you
imagine what we'd sound like if we had to color and spunk-up our expressions
with "hades"? Gee, nobody would know what the **** we're talking about!

There's no end to ****'s practical applicability and adaptibility to general
usage. For instance, you can use it to:
1- give compliments: you're one **** of a woman/man
2- surrender: beats the **** out of me
3- make your point: ...until **** freezes over
4- terminate visits: get the **** out!

Let's for a moment free ourselves of ancestral washing of these brain cells
whose job it is to picture the beyond. Now we can take an impartial look at
**** as an eternal abode, and see one **** of a lot of advantages:
1- no need to jog or lift weights to work up a sweat
2- elimination of heating bills and fire insurance premiums
3- easy to practice walking on hot coals
4- your wife won't bother you for a fur coat.

Let's for a minute longer hang on to this newly-found-impartiality and have
a look at the other eternal abode. Yes, it's been made to sound so much
better, but is it really? Like **** it is...I mean, for how long can the
following be interesting:
1- getting your wings clipped every second saturday
2- meals of fried manna, or manna-a-la-king, or mannaburgers...
3- walking in the park with your lover holding wing-tips
4- doing the hula-hoop with your halo.

Plus see how little this abode adds to everyday sayings. Stuff like "good
heavens!" or "heavens to Betsy!"...how weak. Just try and use it to kick
somebody out: get the heaven out of here. Just don't work...makes you feel
as helpless as a toy balloon in a hurricane.

And how can you terminate a healthy household dispute with such a weak word?
Where would coupled life be if the better half was restricted to wrapping up
with "why don't you go to heaven", the other better half having no choice
but firing back a similar repartee "why don't you go to heaven yourself".
Heavens to Betsy: none of the halves would have the proper stimulus required
to stamp out of the house, or sleep on the couch.

Let's get the **** back to ****. Forgot to mention good behavior. Yes, there
are rewards for good behavior. As example, possibly a transfer to heat control
headquarters. And you can finally get even by turning on the heat real high,
to super-hot, say in the overpopulated math teacher's section!

Think I'll conclude now, before you all wonder If I've temporarily succumbed
to the jug. Conclusions are tricky as (you guessed it) ****, and I sure hope
this one will be to your entire satisfaction. I've decided to sign off with
an unmoving example of what it means to really "feel like ****". You'll find
it in the next and final paragraph.

Suffering from stenosis at any major point along the thirty-foot Via Dolorosa
running from the pylorus to the sigmoid flexure, alleviation in the end only
possible by reaching back for the insertion of small rectified cone-shaped
medicinal objects in gelatinous cases.
 
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