Alan

If it is log to the base 10 then inverse log is 10 to the power.  That is if log₁₀x = a then x = 10^a.

If it is log to the base e then it is e to the power.  If log_ex = a then x = e^a.

In general if log_bx = a then x = b^a is the inverse, where b is any base.

The height of an equilateral triangle with side length x is x*(√3)/2.  Area is (1/2)base*height = (1/2)*x*x*(√3)/2

so Area = (√3)*(x/2)². In your case x = 8√3 so (x/2)² = 16*3 = 48.  therefore Area = 48√3 square inches. 

Wavelength is speed/frequency, so we need the speed of sound in air.  If we assume dry air at sea level this is about 343 m/s.  If original wavelength is L, then L=343/845m so new wavelength, Lnew is

$${\mathtt{Lnew}} = {\frac{{\mathtt{343}}}{\left({\mathtt{2}}{\mathtt{\,\times\,}}{\mathtt{845}}\right)}} \Rightarrow {\mathtt{Lnew}} = {\mathtt{0.202\: \!958\: \!579\: \!881\: \!656\: \!8}}$$  metres

or Lnew ≈ 0.203m

Do you mean for display purposes here?  Just use the subscript option in the Format menu.

log₃(x+1) = 4

Heat absorbed = mass*specific heat*temperature rise

The units of specific heat are important. You didn't specify them, but for water the specific heat is 4.2 J/(g.K).  So the heat absorbed is 10*4.2*(35-25) J = 420 Joules.

Do you mean in the calculator here?  For what purpose: plotting a graph (yes, but graphing here isn't very good)?; finding the roots of the equation (yes, but might depend on the nature of the equation!)?

First rewrite this to isolate y.

7y = 3x - 9

y = (3/7)x - (9/7)

In this form the slope is given by the number multiplying the x, so the slope is 3/7

If there are eight "wheels", four of which have the letters A to Z and four of which have the digits 0 to 9, then there are 26⁴*10⁴ possible combinations, so the probability that a random guess will get the right combination is:

$${\mathtt{p}} = {\frac{{\mathtt{1}}}{\left({{\mathtt{26}}}^{{\mathtt{4}}}{\mathtt{\,\times\,}}{{\mathtt{10}}}^{{\mathtt{4}}}\right)}} \Rightarrow {\mathtt{p}} = {\mathtt{0.000\: \!000\: \!000\: \!218\: \!829\: \!9}}$$

or about 2*10^-10

Of course if a different alphabet is used that has a different number of letters, the probability will be different!

Turn both sides upside down and the equation is still balanced

$$$\frac{x}{50}=\frac{256000}{1350}$$ $

Multiply both sides by 50

$$$x=\frac{50*256000}{1350}$$ $

$${\mathtt{x}} = {\frac{{\mathtt{50}}{\mathtt{\,\times\,}}{\mathtt{256\,000}}}{{\mathtt{1\,350}}}} \Rightarrow {\mathtt{x}} = {\mathtt{9\,481.481\: \!481\: \!481\: \!481\: \!481\: \!5}}$$

 This can also be written as

  $$$x = 9481\frac{13}{27}$$ $

If this is y = (1/4)*sin(x-3) then the derivative is just dy/dx = (1/4)*cos(x-3)