Which value is higher?

Question

1st equation: 5log (x+1) + 10

2nd equation: 10log (x+1) + 5

Answer 1

They're not EQUATIONS if they don't contain an equals sign

Answer 2

Let d = [5log(x+1) + 10] - [10log(x+1) + 5]

= 5 - 5log(x+1)

= 5(1 - log(x+1))

If log(x+1) < 1 (-1 < x < 9) then d > 0, so

5log(x+1)+10 > (10log(x+1)+5)

If log(x+1) = 1 (x = 9) then d = 0, so

5log(x+1)+10 = (10log(x+1)+5)

If log(x+1) > 1 (x > 9) then d < 0, so

5log(x+1)+10 < (10log(x+1)+5)

Answer 3

If we plotted this on a scale of u = log(x+1), there would be two equations as follows:

y = 5u + 10

and

y = 10u + 5

These are equal at u = 1

The second equation has greater slope, so the 1st equation is greater to the left of u=1, and the 2nd equation is greater to the right of that point.

You can now work back to the original equations, bearing in mind that the equations may be undefined for some values of x.

Answer 4

It depends on the value of x.

If x < 9, then the first equation has the higher value.

If x > 9, then the second equation has the higher value.

If x = 9, then they are equal (15).