10.) \( y=\ln (x-1) \)

Did you know that the natural logarithm function, like the one in \( y=\ln(x-1) \), often pops up in finance, science, and growth models? The natural logarithm is the inverse of the exponential function, making it super handy when you need to solve for variables in situations involving growth rates, such as population growth or interest rates! When the input (x) is greater than 1, it allows you to calculate real outputs, which can signify a shift or baseline in a given context. Be cautious with this function, though! A common mistake is plugging in values for x that are less than or equal to 1, which would result in taking the logarithm of zero or a negative number – not permissible in real numbers! Always remember, the domain of \( y=\ln(x-1) \) is strictly \( x > 1 \), so make sure to keep it above that threshold to avoid falling into the trap of undefined values. Happy log-ing!