Dec 4, 2020 · Hence, $$ \log_a (a^x) = x \text { and } a^ {\log_a {x}}=x $$ are both true by definition. The hardest part is trying to explain why these two conceptions of logarithms, while …

Yes one can deduce that $\log a \log b$ is also $\log (b^ {\log a})$. These equations are not mentioned much, perhaps because they can easily be deduced from the other laws (and it …

Jan 9, 2017 · This is called the principal complex logarithm and is usually denoted by $\operatorname {Log}$ (capital L). Technically, it doesn't matter to what range you restrict …

Dec 3, 2025 · You wrote $$\log_5 (\log_4 3) = \frac {\ln 3 - \ln 4} {\ln 5}, \quad \log_6 (\log_6 3) = \frac {\ln 3 - \ln 6} {\ln 6},$$ but actually $$\log_4 3 = \frac {\ln 3} {\ln 4} \quad\text {so}\quad \ln …

Jan 13, 2021 · If you accept (otherwise this can easily be proved) that $\log ()$ is a concave function, then it suffices to show (cf. Jensen) that $x$ is a tangent to $\log (1+x)$.

I thought that $\\log(n!)$ would be $\\Omega(n \\log n )$, but I read somewhere that $\\log(n!) = O(n\\log n)$. Why?

Apr 23, 2017 · I have seen people look at log (several digit number) and rattle off the first couple of digits. I can get the value for small values (aka the popular or easy to know roots), but is …

May 23, 2019 · Log base 4 (7) can be evaluated using the change of base formula. This can similarly be expressed as ln7 divided by ln4, using the natural logarithm. Using the above …

Dec 5, 2023 · Since the default base of log can vary between and even within fields, seems a good rule of thumb is to treat ln as loge (of course), and log as unknown (re: base …

In mathematics, $\log n$ is most often taken to be the natural logarithm. The notation $\ln (x)$ not seen frequently past multivariable calculus, since the logarithm base $10$ finds relatively little …