What are the potential solutions of $\ln \left(x^2-25\right)=0$ ?
To find the solutions to the equation \(\ln(x^2 - 25) = 0\), we first identify the domain of the function. The natural logarithm \(\ln(u)\) is defined only for \(u > 0\). Thus, we require \(x^2 - 25 > 0\), which implies \(x^2 > 25\), or \(|x| > 5\). Any potential solution must satisfy this condition.
The property of the natural logarithm states that \(\ln(u) = 0\) if and only if \(u = 1\). Therefore, we set the argument of the logarithm equal to 1:
\[x^2 - 25 = 1.\]
Adding 25 to both sides gives:
\[x^2 = 26.\]
Taking the square root of both sides of \(x^2 = 26\), we find the solutions:
\[x = \pm \sqrt{26}.\]
We check if these satisfy the domain condition \(x^2 > 25\). Substituting \(x^2 = 26\) into the condition, we get \(26 > 25\), which is true. Thus, both values are valid solutions. Comparing these results with the given options, we identify the correct answer as \(x = \pm \sqrt{26}\).
Logarithms are used in various fields, such as chemistry for calculating pH levels, which is defined as \(pH = -\log_{10}([H^+])\). Similarly, determining the magnitude of earthquakes using the Richter scale involves logarithmic calculations. For example, if you know the amplitude of seismic waves, you can use a logarithmic equation to find the magnitude, much like solving the equation above helps determine the value of a variable within a logarithmic constraint in real-world signal analysis or data modeling.