Question

Solve log(x+1)= -x^2 +10

Answer 1

`\boxed{x \approx 3.064}`

Explanation:

There is no general property that we can use to rewrite:

`log_(a)(u\pm v)`

Then, we'll solve this problem graphically. Let's say that we have two functions:

`f(x)=log(x+1) \\ \\ g(x)=-x^2 +10`

`f(x)` is a logarithmic function translated one unit to the left of the pattern logarithmic function
`log(x)`. On the other hand,
`g(x)` is a parabola that opens downward and whose vertex is
`(0,10)`. So:

`f(x)=g(x)`

implies that we'll find the value (or values) where these two functions intersect. When graphing them, we get that this x-value is:

`\boxed{x=3.064}`

Then, for
`x=3.064`:

`f(x)=log(x+1) \\ \\ f(3.064)=log(3.064+1) \\ \\ f(3.064)=log(4.064) \\ \\ Using \ calculator: \\ \\ f(3.064) \approx 0.6 \\ \\ \\ g(x)= -x^2 +10 \\ \\ g(3.064)= -(3.064)^2 +10 \\ \\ g(3.064)=-9.388+10 \\ \\ g(3.064) \approx -0.6`