VCE Methods Question Thread!

[aspiringantelope]

Hey everyone,
for the changing base formula below, how do you get b? Or is b always 10?
Thanks

[RuiAce]

Hey everyone,
for the changing base formula below, how do you get b? Or is b always 10?
Thanks

\(b\) can actually be any positive number you want, apart from 1. So you could make it 10 if you wanted to, or you could make it 2, or \(e\), or so on.

[aspiringantelope]

\(b\) can actually be any positive number you want, apart from 1. So you could make it 10 if you wanted to, or you could make it 2, or \(e\), or so on.

So it can be any number but it has to be put on both the numerator and denominator, regardless of anything else?

[RuiAce]

So it can be any number but it has to be put on both the numerator and denominator, regardless of anything else?

I'm not sure what you mean by "has to be put on both the numerator and the denominator".

But if you mean the fact that \(b\) appears both in the numerator/denominator and you're putting the same value in for both of them, then well that has to be true.

[aspiringantelope]

I'm not sure what you mean by "has to be put on both the numerator and the denominator".

But if you mean the fact that \(b\) appears both in the numerator/denominator and you're putting the same value in for both of them, then well that has to be true.

Yeah I mean that,
and thanks a lot for the clarification!

[lzxnl]

Hey everyone,
for the changing base formula below, how do you get b? Or is b always 10?
Thanks

It probably helps to know where this formula comes from.

Firstly, let us write out what means. Using the definition as inverse of exponential:

This is all fine and dandy. Now, you want to convert this to the original base, a.  In particular, you want to find an expression for in terms of
This suggests that we need to take a base a log of both sides. Proceeding with log laws gives

Surprised? In words, the formula reads

Look at the relative positions of the old and new bases on both sides. See how the old base is used in the quotient.

This illustrates one thing you must learn about mathematics. It's NOT about remembering formulas. Now, it turns out that I've used change of base enough times to be able to remember it. However, I never actively committed it to memory. I just understood how the proof worked, and I never had to remember a thing. Try go through the reasoning behind this derivation.

[aspiringantelope]

It probably helps to know where this formula comes from.

Firstly, let us write out what means. Using the definition as inverse of exponential:

This is all fine and dandy. Now, you want to convert this to the original base, a.  In particular, you want to find an expression for in terms of
This suggests that we need to take a base a log of both sides. Proceeding with log laws gives

Surprised? In words, the formula reads

Look at the relative positions of the old and new bases on both sides. See how the old base is used in the quotient.

This illustrates one thing you must learn about mathematics. It's NOT about remembering formulas. Now, it turns out that I've used change of base enough times to be able to remember it. However, I never actively committed it to memory. I just understood how the proof worked, and I never had to remember a thing. Try go through the reasoning behind this derivation.

Wow. Didn't expect such a deep comprehension but wow.Thanks a lot!

[lzxnl]

Wow. Didn't expect such a deep comprehension but wow.Thanks a lot!

No problem! This is the level of understanding you want to have for every topic. If you can step yourself through every step and understand the point of each step, you're doing well.

[leo_has_atarnotes]

Hello Forum!

For this equation:

e^ln(9)=9

I'm unsure why the e's cancel out.

I feel like I'm missing something very obvious about this. Any suggestions welcome!

 

[AlphaZero]

Hello Forum!

For this equation:

e^ln(9)=9

I'm unsure why the e's cancel out.

I feel like I'm missing something very obvious about this. Any suggestions welcome!

Well, it might help to look at the definition of a logarithm to note that the logarithm is the inverse operation of an exponent. 

\(x=\log_a(b)\)  by definition means  \(a^x=b\). Or in other words,  \(\log_a(b)\) tells us what power we need to raise \(a\) by to get \(b\).

So, in your example,  \(\log_e(9)\) is the number we need to raise \(e\) by to get \(9\),  and so  \(e^{\log_e(9)}=9\).

To generalise,  \(a^{\log_a(b)}=b\)  and  \(\log_c(c^d)=d\).

[leo_has_atarnotes]

Well, it might help to look at the definition of a logarithm to note that the logarithm is the inverse operation of an exponent. 

\(x=\log_a(b)\)  by definition means  \(a^x=b\). Or in other words,  \(\log_a(b)\) tells us what power we need to raise \(a\) by to get \(b\).

So, in your example,  \(\log_e(9)\) is the number we need to raise \(e\) by to get \(9\),  and so  \(e^{\log_e(9)}=9\).

To generalise,  \(a^{\log_a(b)}=b\)  and  \(\log_c(c^d)=d\).

Thank you! That certainly clears things up.

[Scribe]

Hey all,

I noticed in the Cambridge Mathematical Methods Unit 3&4 textbook that there is an 'Appendix A: Counting methods and the binomial theorem' section. Is this assumed knowledge and do we need to learn this for the course? I remember going over it in Units 1&2 very briefly.

Thanks

[Sine]

Hey all,

I noticed in the Cambridge Mathematical Methods Unit 3&4 textbook that there is an 'Appendix A: Counting methods and the binomial theorem' section. Is this assumed knowledge and do we need to learn this for the course? I remember going over it in Units 1&2 very briefly.

Thanks

It is unlikely that it comes up in the current study design and I believe that it is rarely taught as a part of the course.

However, it doesnt hurt to have a decent understanding and  learn it just in case (from memory it has come up in past study designs e.g. 2000-2005)

[S_R_K]

You can be asked to calculate nCr values by hand in Exam 1 if there is a question involving the binomial distribution.

You should know (1) the definition of nCr in terms n! and r!, and (2) some simple identities / shortcuts (ie. that nCr = nC(n–r), that nCn = 1, etc.)

[TheAspiringDoc]

Hey all,

I noticed in the Cambridge Mathematical Methods Unit 3&4 textbook that there is an 'Appendix A: Counting methods and the binomial theorem' section. Is this assumed knowledge and do we need to learn this for the course? I remember going over it in Units 1&2 very briefly.

Thanks

Also, sometimes schools don't go exactly by the study design (I had hypergeometric probability in one of my SACs this year :O ) so it can often be good to ask the 'could this be relevant?' questions to your teacher if it's to prepare for SACs