Question: yeah i was asking if it was illicit since my textbook has a way to do by first changing it all into base $\ln$ natural logarithm
1. The problem is to determine if it is valid (licit) to change logarithms of any base into natural logarithms (base $e$) before solving.
2. The key formula used is the change of base formula for logarithms:
$$\log_a b = \frac{\ln b}{\ln a}$$
This formula states that the logarithm of $b$ with base $a$ can be expressed as the ratio of natural logarithms of $b$ and $a$.
3. This is always valid because $\ln$ is just a logarithm with a fixed base $e$, and the ratio converts between bases.
4. For example, to compute $\log_2 8$, you can write:
$$\log_2 8 = \frac{\ln 8}{\ln 2}$$
5. Evaluating the right side:
$$\ln 8 = \ln (2^3) = 3 \ln 2$$
So:
$$\log_2 8 = \frac{3 \ln 2}{\ln 2}$$
6. Canceling $\ln 2$:
$$\frac{3 \cancel{\ln 2}}{\cancel{\ln 2}} = 3$$
7. This matches the known value of $\log_2 8 = 3$, confirming the method is correct.
8. Therefore, it is not illicit but perfectly valid and often convenient to convert logarithms to natural logarithms using the change of base formula before solving.