Maths C3 - Logs and Exponentials... HELP???

Yeah sorry. I'm normally really good at explaining what it is that I'm specifically struggling with.

It's this part that I don't really understand
...

"...Then $2^{\left(\log_2 8\right)}$ means "2 to the power of the thing you need to raise 2 by to get 8".

If you think about it, this is just going to take you back to 8 which means that $2^{\left(\log_2 8\right)}=8$. There is some "cancelling" going on here since $2^x$ and $\log_2 x$ are inverse functions.

Of course a similar principle will help you understand why $e^{\log_e x} = x$ "

It's good that you're trying to understand this but don't spend too long thinking about it! In the exam you can quote $e^{\ln x} = x$ without explanation/proof and I think you already understand the proof I gave you using log laws.

I'll show you an alternative thought process below but I still feel the explanation in my previous post was better and I can't really explain it more. So here's another way to think of it:

$\log_2 8$

This is the power you need to raise 2 by to get 8. So $\log_2 8$ is the solution of the equation

$2^x=8$

So here $x$ is $\log_2 8$.

Then $2^{\log_2 8}$ is the same as $2^x$ which is 8 using the equation above.

So I'm having a dull moment yet again. Can someone tell me whether there is a difference between the following?...


$ln (3)^x$

and...
$ln (3^x)$

Is it even possible to use the "log power rule" on the second one to give?...
$x ln (3)$ ??

There is no difference. And yes.

Yes there is.

E.g. $\log (3)^2 = \log 3 \times \log 3$

$\log(3^2) = 2\log 3$


@Philip-flop You can only use the log law for $\log(a^x)$.

$\log(a)^x$ is a different thing.

From what I've seen, $\log(a)\log(a)=\log^2(a)$ just like $\sin(x)\sin(x)=\sin^2(x)$

Then $\log(a)^n$ just refers to the entire thing inside the log being raised to $n$, which is a useful notation if $a$ is an long or complicated expression.

Yes I've seen the notation $\log^a n$ used which is equivalent to $\log(n)^a$ just like $\sin^a n$ is equivalent to $\sin(n)^a$.

In A Level $\log(n)^a$ would mean $(\log(n))^a$. Are you sure you've seen it mean $\log(n^a)$? This doesn't feel right.

Yes there is.

E.g. $\log (3)^2 = \log 3 \times \log 3$

$\log(3^2) = 2\log 3$


@Philip-flop You can only use the log law for $\log(a^x)$.

$\log(a)^x$ is a different thing.

Yes that was what I had come across before! Couldn't remember which one you could use the log power rule on!

@RDKGames and @notnek but now I'm confused between..
$log^2 (a), log(a)^2, [log(a)]^2$ to me these are all the same but the second one is debatable and can be misleading!

Normally $\log(a)^2 = (\log (a))^2$ and $\sin(a)^2 = (\sin (a))^2$ and you'll see it like that here for example.

But it can be ambiguous so you probably wouldn't see it like that in an exam without more brackets.

Is this all for gcse level?!

No this is all A Level.

Is this all for gcse level?!