Verified answer

The second one is easier than the first. Is the 40 meant to be a subscript? I assume so.

I write log base 40 as log[40] so the equation is

log[40](x+7) + log[40](6-x) = 1.

Now use the log property that in any base,

log a + log b = log ab

Therefore the left side is

log[40]((x+7)(6 - x))

which is equal to

log[40](42 - x - x²)

Since this is equal to 1,

42 - x - x² = 40

x² + x - 2 = 0

(x + 2)(x - 1) = 0

x = -2 or 1.

Sorry, can't get a handle on the first one, except to notice that x=0 satisfies it because

3^0 - 1 + 5^0

= 1 - 1 + 1

= 1

and 2^0 = 1.

I suspect there aren't any other solutions, but can't prove it.