To calculate logarithms with a specific base on a TI-89 Titanium, the change of base formula is used: ln(value) / ln(base). This method allows for the calculation of logarithms in any base, including base 10 and natural logarithm. For example, ln(2)/ln(10) gives the log of 2 in base 10, while ln(2) remains unchanged for base e. For calculating roots, the zeros() function is recommended, using the syntax zeros(expression, variable) or solve(equation, variable). If the root function is not working as expected, it may be due to incorrect settings, such as language preferences. Additionally, users can utilize built-in functions like log(number, base) for simpler calculations, especially if they have upgraded features through apps like Omnicalc. The log function can also be accessed quickly by pressing the Diamond button followed by 7.