Solve for x. Don't do the work using algebra and don't use a calculator. The Laws of Logarithms will be helpful.
ln(x^2 - 1)= 3
To solve for x in the equation ln(x^2 - 1) = 3, we can use the inverse property of logarithms, which states that if ln(a) = b, then e^b = a. Here, e refers to Euler's number, approximately 2.71828.
1. Begin by rewriting the equation using the inverse property of logarithms:
e^3 = x^2 - 1
2. Simplify e^3:
e^3 is approximately equal to 20.08554
3. Rewrite the equation with the simplified value:
20.08554 = x^2 - 1
4. Rearrange the equation to isolate x^2:
x^2 = 20.08554 + 1
x^2 = 21.08554
5. Take the square root of both sides to solve for x:
x = ±√21.08554
Therefore, the solutions for x are approximately x = ±4.6.