Application of simple differentiation

Question 1
Water is being drained from a pond such that the volume V (in m^3) of water in the pond after t hours is given by V = 5000(60 - t)^2. Find the rate at which the pond is being drained after 4 h.

Question 2The velocity of an object moving with constant acceleration can be found from the equation v = (v[0] ^2 + 2as)^2, where v[0] is the initial velocity, a is the acceleration, and s is the distance traveled. Find dv/ds.

Example 3The electric field E at a distance r from a point charge is E = k/r^2, where k is a constant. Find an expression for the instantaneous rate of change of the electric field with respect to r.

Example 4The distance s (in m) traveled by a subway train after the brakes are applied is given by s = 20t - 2t^2. How far does it travel, after the brakes are applied, in coming to a stop?

Look at the following and made some comments to assist.
Remember the log base is in [] brackets

Example 1
log [8] (x + 2) = 2 - log [8] 2
log [8] 1 + log [8] 2 = 2 / (x + 2)

Example 2
log [8] (x + 2) = 2 - log [8] 2
log [8] (x + 2) + log [8] 2 = 2
log (2x + 4) = 2
log (2x + 8) = 2
log 8
2x + 4 = 1.8
2x = 1.8 - 4

Example 3
log [8] (x + 2) = 2 - log [8] 2
log [8] x + log [8] 2 = 2 - log [8] 2
log [8] x + 2 log [8] 2 = 2

Example 4
log [8] (x + 2) = 2 - log [8] 2
log [8] (x + 2) = 1.67
log [8] 1.67 = x + 2

Example 5
log [8] (x + 2) = 2 - log [8] 2
log [8] (x + 2) + log [8] 2 = 2
log [8] 2(x + 2) = 2
log [8] (2x + 4) = 2
log [8] 2x = 2 - 4

Example 6
log [8] (x + 2) = 2 - log [8] 2
log (x + 2) / log 8 = 2 - log 2 / log 8
log (x + 2) / 0.903 = 2 - 0.33
log (x + 2) 0.903 * 1.667
log x + 2 = 1.5
log x = 1.5 - 2
log x = 0.5
x = 0.32

Simple logs

Examine the following statements.
Remember that the base will be represented in (). Help these students.

Example 1
log (4) x = 12
log x = log 12
log 4
log x = 1.08 * 0.6
log x = 0.65
x = 4.5

Example 2
log (4) x = 12
log 4 = 12
x = 12 / log 4
x = 20

Example 3
log (4) x = 12
log (4) x = log 12

Example 4
log (4) x = 12
log (4)12 = x

Example 5
log (8) 32 = x
x = 8/32
x = 1/4

The [] is used since subscript and superscript are not allowed.
2 [3] = 8 can also be expressed as log [2] 8 = 3
Is this correct?
0.035 [x] = 2.74
Find x?

Re-express as a log which is log [0.035] 2.74 = x

1. Draw line
2. log 2.74__ = x
   log 0.035

Or is this correct?
0.035 [x] = 2.74 Find x?

1. Log both sides
2. log 0.035 [x] = log 2.74
3. Re-express to remove power
   x log 0.035 = log 2.74
4. Make x the subject of the equation
   x = log 2.74 / log 0.035

Logs

What really is logs?
Do you think this is an important aspect of maths?
How is logs related to exp?
What is the basic strategy in logs?

Logs

What really is logs?
Do you think this is an important aspect of maths?
How is logs related to exp?
What is the basic strategy in logs?

Exponent

What is the purpose of exponent?
Is a quadratic an exponent?
It is said that everything in life involves some aspect of Maths, give some real life scenarios thanvolves some aspect of Maths, give some real life scenarios that involves the exponent aspect of maths.