You are Here: Home >< Maths

# P2 Logs Question watch

1. i) show that log2w = 2log4w

The above I know how to do, but:

ii) Hence solve the simultaneous equations:

log4t² - log2U² = (log0.59) - 1

4^t = 2^(U-2)

where U>0 and t>0

2. First we convert log[0.5](9) to base-2. (You can use a standard formula here if you prefer.) Let x = log[0.5](9). Then 0.5^x = 9, so 1 = 9*2^x, so 0 = log[2](9) + x. So log[0.5](9) = -log[2](9)

Now we simplify the more complicated equation:

log[4](t^2) - log[2](U^2) = log[0.5](9) - 1
(1/2)log[2](t^2) - log[2](U^2) = -log[2](9) - 1 . . . . . converting everything to base-2
log[2](t) - log[2](U^2) = -log[2](9) - 1
log[2](U^2) - log[2](t) - log[2](9) = 1
log[2](U^2/(9t)) = 1
U^2/(9t) = 2
U^2 = 18t

Since also 4^t = 2^(U - 2),

2^(2t) = 2^(U - 2)
2t = U - 2
(1/9)U^2 = U - 2
U^2 - 9U + 18 = 0
(U - 6)(U - 3) = 0
U = 3 or 6

So the solutions to the given equations are:

(A) U = 3, t = 1/2;
(B) U = 6, t = 2.

That was quite hard.

### Related university courses

TSR Support Team

We have a brilliant team of more than 60 Support Team members looking after discussions on The Student Room, helping to make it a fun, safe and useful place to hang out.

This forum is supported by:
Updated: June 20, 2005
The home of Results and Clearing

### 2,009

people online now

### 1,567,000

students helped last year
Today on TSR

### University open days

1. Keele University
Sun, 19 Aug '18
2. University of Melbourne
Sun, 19 Aug '18
3. Sheffield Hallam University
Tue, 21 Aug '18
Poll
Useful resources

### Maths Forum posting guidelines

Not sure where to post? Read the updated guidelines here

### How to use LaTex

Writing equations the easy way

### Study habits of A* students

Top tips from students who have already aced their exams