The main difference is that when computing limits using logs, the logarithmic function is used, whereas when computing limits using powers, the power function is used. This means that the approach and method of calculation may differ. Additionally, the values of the limits may also differ depending on which method is used.
Logs are useful for computing limits when the function being evaluated has a complex or indeterminate form, such as 0/0 or infinity/infinity. In these cases, taking the logarithm of the function may simplify the expression and make it easier to evaluate the limit.
No, logs may not always be the best approach for computing limits. In some cases, using powers may be more efficient and yield a more accurate result. It is important to understand the properties and limitations of both logs and powers when deciding which method to use for computing a limit.
Choosing between logs and powers for computing a limit depends on the specific function being evaluated and its properties. Generally, if the function has a logarithmic form, then using logs may be a good approach. Otherwise, using powers may be more suitable.
One common mistake is forgetting to apply the properties of logs or powers when evaluating a limit. It is important to have a thorough understanding of these properties and how they affect the limit calculation. Another mistake is relying solely on one method (logs or powers) and not considering alternative approaches, which may be more efficient or accurate.