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I have no idea what to do at all, the book gives no relevant examples so I had to give up.

I am suppose to find $\sinh(\ln 2)$ I don't even know where to start. I think I need to use the form $\dfrac{e^{\ln2}-e^{-\ln2}}2$ but I get confused on how to do that as well.

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Just use the fact that $e^{\ln x}=x$ for all $x>0$ and $e^{-x}=\frac 1{e^x}$ for all $x\in\mathbb R$. – Davide Giraudo Oct 6 '11 at 20:07
Recall that logarithm is the inverse of exponentiation, i.e. for $\mathrm{e}^x = 2$, the real solution is $x= \ln 2$, now substitute the solution back into the equation (lightbulb) – Sasha Oct 6 '11 at 20:09
I know I should never ask why, but is there an easy way to remember that? – user138246 Oct 6 '11 at 20:09
"The logarithm is the inverse of the exponential" is often the definition of the logarithm. – Arturo Magidin Oct 6 '11 at 20:10
So e raised to the ln2 just means e to the $loge^2$ – user138246 Oct 6 '11 at 20:11
up vote 4 down vote accepted

As I've mentioned before, trying to pass calculus through sheer memorization effort is doom to fail.

Here you are asking of a way to memorize the fact that the $$e^{\ln 2} = 2.$$ This is simply a consequence of what the logarithm function is supposed to be.

Logarithms are the inverses of exponentials. Inverse functions are functions that "undo" what the "original" function does. The logarithm base $a$ "undoes" exponentiation base $a$; exponentiation base $a$ "undoes" the logarithm base $a$. We define $\log_a(b)$ to be the number $r$ exactly when $a^r = b$. The statement "$\log_a(b)=r$" and the statement "$a^r = b$" mean the same thing, just like the statement "$\frac{6}{2}=3$" and the statement "$3\times 2= 6$" mean the same thing (because "$\frac{6}{2}$" means "the number that multiplied by $2$ gives $6$").

That is, we always have $$ a^{\log_a(x)} = x\quad\text{and}\quad \log_a(a^x) = x\quad\text{for all values of }x$$

These identities are simply a consequence of the fact that each function "undoes" the other function. It is a waste of mental effort to try to memorize them: there's just too many of them! One for every possible value of $a$. It's far better (and more productive in the long run) to understand why they are true, so that they can be internalized as conclusions rather than memorized "magic spells" that have no meaning.

In particular, since $\ln x$ is the logarithm base $e$ of $x$, $\ln x = \log_e(x)$, the fact that the exponential undoes the logarithm and the logarithm undoes the exponential means that $$e^{\ln(x)} = x\quad\text{and}\quad \ln(e^x) = x\quad\text{for all values of }x.$$

So $e^{\ln 5} = 5$, $\ln(e^3) = 3$, $e^{\ln\pi} = \pi$, $17^{\log_{17}(42.4)} = 42.4$, and so on and so forth.

Since we also have that, among the properties of logarithms, $r\ln(b) = \ln(b^r)$,

So: $$\sinh(\ln(2)) = \frac{e^{\ln 2} - e^{-\ln 2}}{2} = \frac{e^{\ln 2} - e^{\ln(2^{-1})}}{2} = \frac{2 - 2^{-1}}{2} = \frac{\quad 2 - \frac{1}{2}\quad}{2} = \frac{2}{2} - \frac{\;\frac{1}{2}\;}{2} = 1 - \frac{1}{4} = \frac{3}{4}.$$

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I still don't get $a^{\log_a(x)} = x\quad\text{and}\quad \log_a(a^x) = x\quad\text{for all values of }x$ I know I am stupid but I just can't get it for some reason. If I have 2 raised to a number that is suppose to make a number x than how do I know that equals x? That doesn't make sense to me, to me it could be anything. – user138246 Oct 6 '11 at 21:53
The reason is that $\log_a(x)$ is defined to be the number to which you need to raise $a$ in order to get $x$ as the answer. You are raising $a$ to the number which is supposed to give you $x$ as the answer; how do you know that $x$ is the answer? Because that's what the number you are raising $a$ to is supposed to do. – Arturo Magidin Oct 6 '11 at 21:56
So I need a to equal x by using an exponent. so $a^x$=x? I don't get that. If I replace a with 3 and x with 9 I get x don't I? So shouldn't it be equal to a? – user138246 Oct 6 '11 at 21:58
$\log_a(x)$ is the only thing you can put in the empty box in $$a^{\Box} = x$$to make that equation true. There is one, and only one number that can fill that box and make the equation true. That number is what $\log_a(x)$ is defined to be equal to. – Arturo Magidin Oct 6 '11 at 22:01
So I need to make a equal to x by raising it by an exponent, I don't understand how raising it by x makes a x. Like if I am trying to make 4 = 16 I raise it by 2 not 16. – user138246 Oct 6 '11 at 22:03

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