log1/4[(1/4)-2x] = log1/4(64)
-2x = log1/4 64
-2x = log10(64)/log10(1/4) = loge(64)/loge(1/4) = -3
Thus x = 3/2 = 1.5
In fact we can check that (1/4)-2(3/2) = (1/4)-3 = 43 = 64
Remember that to find logaB for any base a, we can always use the logs...

I misunderstood the problem. Sorry.

Well... 10,000 cannot be right. In fact,
log(log(10,000) = log(4) .
Think of it like this:
10log(log(x)) = 104
which is the same as:
log(x) = 104
And therefore x = 10^104 which is a huge number: 1010...

It would have been better if you had shown us what you have done so far. Instead, it seems to me that you are just trying to get your assignment/homework done by the tutors.
How does one solve a system of equations by substitution? We can solve the first equation with
respect...

3
∫ ---------- dx ?
√(4-x2)
but what are the limits of integration? Forget the 3 for a moment, just take the constant out,
and write the denominator as
√4(1-x2/4) = 2√(1-x2/4)
let...

With this kind of problems, one never goes wrong by counting the number of possible outcomes and the number of favorable outcomes.
FAVORABLE OUTCOMES: (2,6), (3,5), (4,4), (5,3), (6,2) ... these sum up to 8.
...

If you use log() for logarithm in base 10, then from
log(x+3) = 3
to pass to an exponential form we do the following:
10log(x+3) = 103 = 1000 (*).
Now, using properties of log's, it follows that
10log(x+3)...

It is nothing else but a quadratic equation:
- 4cos2(x) +4√3cos(x) - 3 =0
multiply by -1 both sides to yield
4cos2(x) -4√3cos(x) +3 =0
Now let z = cos(x) and you get the equation
4z2 - 4√3z +3 = 0 ...

Call A the slower car and B the faster car.
Let x be the speed of B
Then, according to the text of the problem, the speed of A is x-10.
Since the cars are traveling toward each other, when they meet the sum of the distances each car has traveled is
400. So, the problem can be solved...

One simple approach is to consider that there are 2 grey males (M1, M2) and 5 grey females (F1, F2, .. , F5).
If you form all grey couples with these 7 mice, the possibilities are
M1M2, M1F1, M1F2, M1F3, M1F4, M1F5, M2F1, M2F2, M2F3, M2F4, M2F5 i.e, 11...

I am sure that your textbook describes this topic very well. All it takes is basically some reading.
In any case, the z-score is found as
z-score = (60-50)/6 = 10/6

This topic deals with implicit differentiation for which I am sure your textbook has a lot to say.
I assume that you are expected to find y'(x) = dy/dx since nothing is specified in the text of the problem and that is usually what is done, i.e., we are working as if y=y(x).
Look...

Well, if you invest K today, and the annual interest is r, then using simple interest, at time t, your investment is worth
A = K(1+r*t).
We want A = 4K and r=5%. Thus:
4K = K(1+.05*t)
or
4 = (1+.05*t)
which...

By definition the rate of change is
f(c+h)-f(c) [-(c+h)^2+2(c+h)-3] - [- c^2 +2c-3]
------------- = ----------------------------------------------
c+h - c ...

(3.7-1.2)=c/2 In fact we knew that before adding 1.2 gallons the bucket was half full (c/2) and, clearly, if now we have 3.7 gallons after adding extra 1.2, before adding there were (3.7-1.2)=2.5 gallons of water in the bucket.
Hence,
c = 2*(3.7-1.2)=2*2.5 = 5, i.e. the...

To find the domain of the function assigned, the natural log, you need to specify for which values of x, the function is well defined. We know that for logarithms in general, natural or in any base, we need the argument to be positive, i.e.,
x^2 - 3x - 4 > 0
Now...

Ana,
The domain of definition of a function is the set of values of x for which the assigned function exists. In this case you are given a ratio of two second degree polynomials. However, if you factor both numerator and denominator, you get
...

Variables are
m = number of cars;
n = number of buses;
max: Profit = 2.50*m +7.50*n
s.t.: 6*m+30*n <= 600 (we can use 60% of the total 20x50=1000 sqm parking lot)
n+m <= 60 ...

The easiest, and perhaps the only way, to solve this kind of problems is to write
y = g(x) = x+1
and solve this equation as a function of y, i.e.,
x = y-1.
Therefore, the inverse function you are looking for is given by g-1(x) = x-1...

Let x be the number of rows.
Leo plants x+29 trees in each row. Hence the number of rows is given by
912 x
------ =
x+29
i.e., after simple algebraic manipulations, 912=...