91Ó°ÊÓ

The natural logarithm is denoted as \( \ln \) and has a base of \( e \), which is an irrational and transcendental number approximately equal to 2.71828. It is particularly useful in solving exponential equations where the exponential term has \( e \) as the base.

In our exercise, after isolating the term \( e^{x^{2}-1} \), taking the natural logarithm on both sides becomes a key step. This is because the natural logarithm function, \( \ln \), is the inverse of the exponential function \( e^{x} \).

By applying \( \ln \) to \( e^{x^{2}-1} = 7 \), we get \( \ln(e^{x^{2}-1}) = \ln(7) \). This effectively "cancels out" the \( e \) on the left side, leaving us with \( x^{2} - 1 \), and thus simplifying the problem significantly.

Logarithms come with various properties that simplify complex calculations. One of the most beneficial properties is the power rule: \( \ln(e^{a}) = a \).

This property is pivotal in our step-by-step solution where \( \ln(e^{x^{2}-1}) = \ln(7) \). It simplifies to \( x^{2} - 1 = \ln(7) \), allowing us to easily handle the expression involving \( x \).

• **Product Rule**: \( \ln(AB) = \ln(A) + \ln(B) \)
• **Quotient Rule**: \( \ln\left(\frac{A}{B}\right) = \ln(A) - \ln(B) \)
• **Power Rule**: \( \ln(A^{b}) = b \cdot \ln(A) \)

These rules are powerful tools in algebra and calculus. They are important for transforming and simplifying equations that involve logarithms.

Rounding numbers is a mathematical method used to simplify figures to make them easier to work with. It is especially important in ensuring that results reflect the precision required by a particular problem.

When rounding to three decimal places, we follow these general steps:

• Identify the digit at the third decimal place.
• Look at the digit immediately to the right; if it is 5 or greater, increase the third decimal place by 1.
• If it is less than 5, keep the third decimal place unchanged.

In our exercise, after determining the solutions for \( x \), we rounded the values to get \( x \approx \pm 2.807 \), providing clarity while adhering to the problem's instructions.