CPhill

1) lne  = 1

2) lne²  = 2 ln e    =  2 (1)  = 2

3) lne¹⁰ = 10 ln e   =  10 (1)  = 10

4) 10 lne   =  10

5) ln1   =  0

6) lne/4  = 1/ 4

7) lne²/2    =  2 ln e  / 2   =  ln e

8) lne⁸³  =  83 ln e    =  83 (1)  =  83

Correct....!!!

No doubt....A math question you're allowed to answer only if you have CIA clearance

Correct...( to the nearest degree...)

Arc length  =      2pi (radius) (x / 360) 

Where x is the angle(in degrees) we're looking for

So  we have

15 pi  =  2pi (60) (x /360)     divide pi out and simplify

15  =  2 (60) * x /360      multiply both sides by 360

5400  = 120 x     divde both sides by 120

45°  =  x

1) 4log₃2 - 2log₃x = 1

Note that  1   =  log₃ 3  .....so we can write

4 log₃ 2 - 2log₃x  = log₃ 3      {  log property :  b log a  =  loga^b  }

log₃ 2^4  - log₃x^2  = log₃3   { log a - log b  =  log (a/b) }   ⇒ 2⁴  = 16

log₃ (16/ x^2)  = log₃ 3

Since  we have equal logs   we can just solve

16/x^2  =  3

x^2  = 16 / 3         take the positive root 

x  =  4 / √ 3   ≈ 2.3094

2) logx² = 2       ....we can write

2 log x  =  2         divide both sides by 2

log x  =  1

This says that    10^1  =  x  =  10

y^2  =  3x + 10

Subtract 10  from both sides

3x  =  y^2 - 10      divide both sides by 3

x  =  (1/3)y^2  - 10/3

This is a  "horizontal" parabola that opens to the right....

I'm sure there is....but....I'm not a good enough geometer to figure it out....

This  is a circle centered at the origin  with a radius  of  9

Total number of gallons in the pool

Total number of minutes filling the pool