WEBVTT

1
00:00:00.000 --> 00:00:03.947
Here is the problem: ln(x squared) equals ln(9).

2
00:00:04.397 --> 00:00:09.970
Solve for x numerically (the equation contains functions such as sin, log or roots).

3
00:00:10.770 --> 00:00:12.268
Start with the equation.

4
00:00:12.718 --> 00:00:14.889
Rewrite as f(x) equals 0.

5
00:00:15.339 --> 00:00:31.291
This equation is not polynomial, so solve it numerically. Scan x from negative 10 to the power 9 to 10 to the power 9 for sign changes of f and for points where the graph touches the axis, then narrow each one down. Every root is checked by substituting it back.

6
00:00:31.741 --> 00:00:36.745
Found 2 solutions in [−10 to the power 9, 10 to the power 9].

7
00:00:37.545 --> 00:00:41.759
So the answer is x equals negative 3, x equals 3.

8
00:00:42.209 --> 00:00:50.975
Values marked approximately are numerical approximations accurate to about 10 significant digits. Every value was checked by substitution.

9
00:00:51.774 --> 00:01:04.951
x equals negative 3: left side equals 2.1972246, right side equals 2.1972246, which checks out.

10
00:01:05.401 --> 00:01:18.509
x equals 3: left side equals 2.1972246, right side equals 2.1972246, which checks out.