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11) a) Show that

log_43=log_2(\sqrt{3})

b) Hence or otherwise solve the simultaneous equations:

2log_2y=log_43+log_2x

3^y=9^x

x,y > 0

I've done part a, but not sure how to start part b.

Reply 1

ghostwalker

ViralRiver

11) a) Show that

log_43=log_2(\sqrt{3})

b) Hence or otherwise solve the simultaneous equations:

2log_2y=log_43+log_2x

3^y=9^x

x,y > 0

I've done part a, but not sure how to start part b.

I'd start with your second equation. From what you know of the laws of indices you should be able to get y in terms of x.

Reply 2

ViralRiver

I can derive

ylog3=xlog9

Reply 3

ghostwalker

ViralRiver

I can derive

ylog3=xlog9

Well, what's 9 in terms of 3?

Reply 4

ViralRiver

3^2, but I don't see how that helps >< .

OOOHH!! y = 2x

Reply 5

ghostwalker

ViralRiver

3^2, but I don't see how that helps >< .

OOOHH!! y = 2x

Grin! Now sub into your first equation and use the hint from a)

Reply 6

ViralRiver

Ok, so I now have

2log_2(2x)-log_2(x)=log_2(\sqrt{3})

I'm not sure how to put everything in terms of x though.

Reply 7

ghostwalker

ViralRiver

Ok, so I now have

2log_2(2x)-log_2(x)=log_2(\sqrt{3})

I'm not sure how to put everything in terms of x though.

Using the laws of logs, try and combine the left hand side into one log function.

a log x = log (x^a)

log b - log a = log (b/a)

Reply 8

clad in armour

ViralRiver

Ok, so I now have

2log_2(2x)-log_2(x)=log_2(\sqrt{3})

I'm not sure how to put everything in terms of x though.

do what the person above just said but be careful with the 2 in front of the first term, youll have to take it up as a power

Reply 9

ViralRiver

Ahh ok, doing that I have

x=\frac{\sqrt{3}}{4}

Reply 10

ghostwalker

ViralRiver

Ahh ok, doing that I have

x=\frac{\sqrt{3}}{4}

Why the question mark. Have faith, you got it right. You just need y now.

Reply 11

ViralRiver

Ahh yes, got y and they're both right. I'm just stuck on the last question now.

12). a) Given that

3 + 2log_2x=log_2y

, show that

y=8x^2

.

Do I need to turn 3 into a logarithm, or is that unnecessary?

Reply 12

ghostwalker

ViralRiver

Ahh yes, got y and they're both right. I'm just stuck on the last question now.

12). a) Given that

3 + 2log_2x=log_2y

, show that

y=8x^2

.

Do I need to turn 3 into a logarithm, or is that unnecessary?

There's no simple yes/no answer to that one. Do it, and see where you get (it will work it).

Reply 13

ViralRiver

That's the thing, I'm not sure how to turn it into one >< . Because

3 \not= log3

Reply 14

ghostwalker

ViralRiver

That's the thing, I'm not sure how to turn it into one >< . Because

3 \not= log3

You can write 3 as

3\times\log_22

since

\log_22

is equal to 1. And use that as the basis for conversion, using the rules previously stated.

Reply 15

ViralRiver

Thanks ! I've just worked out parts b c and d from that so I must be getting the hang of it :P .

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