Question

Graph the system. Use the graph to approximate the solution of the equation
log(x + 1) = –x2 + 10.

es002-1.jpg

Answer 1

I need to know which base “log” uses. Common choices:

- log = log10 (common logarithm)
- log = ln (natural log)

I’ll give both answers.

1) If log means log10:
Solve log10(x+1) = −x^2 + 10.
Graph y1 = log10(x+1) and y2 = −x^2 + 10; they intersect once (domain x>−1) at approximately
x ≈ 3.064. The common y-value there is y ≈ log10(4.064) ≈ 0.61 (and −(3.064)^2 + 10 ≈ 0.61).

2) If log means ln:
Solve ln(x+1) = −x^2 + 10.
Graph y1 = ln(x+1) and y2 = −x^2 + 10; they intersect once (x>−1) at approximately
x ≈ 2.94 (≈ 2.936). The y-value is y ≈ ln(3.936) ≈ 1.37.

How to get this from the graph: plot both functions for x > −1 and find their intersection point (use a graphing calculator or zoom in on the crossing to read x). If you tell me which log you mean, I’ll refine the numeric value further.