Find $x$ for $4^{x-4} = 7$.
Answer I got, using log, was ${\log(7)\over 2\log(2)} + 4$
but the actual answer was ${\ln(7)\over2\ln(2)} + 4$
I plugged both in my calculator and turns out both are the equivalent value.
Anyways, is using either one of ln or log appropriate for this question? Obviously ln is when log has the base e, and log is when it has the base 10.
Final question: How do I know when to use which? that is which of ln or log is used when solving a question??
For example, if a question asks to find $x$ for $e^x = 100$, I will use $\ln$ since $\ln(e)$ cancels out.
If a question asks to find $2^x = 64$, i will use log since "$e$" isn't present in the question.
So is using either $\log$ or $\ln$ the same?
Best Answer
You can use any logarithm you want.
As a result of the base change formula $$\log_2(7) = \frac{\log(7)}{\log(2)} = \frac{\ln(7)}{\ln(2)} = \frac{\log_b(7)}{\log_b(2)}$$ so as long as both logs have the same base, their ratio will be the same, regardless of the chosen base (as long as $b > 0, b\neq 1$).