Logarithm in complex plane (Spot my mistake)

Question

Please spot my mistake,

$$ \begin{align} 0&=0\\ \log(1)&=\log(1)\\ \log((-1)^2)&=\log(1)\\ 2\log(-1)&=\log(1)\\ 2\pi i&=0 \end{align} $$

Actually somestimes i confused with $\log$ and $\ln$. What is the difference? Is $\log$ a logarithm with base of $10$ or $e$?

Answer 1

For real numbers $x$,

1. $\log(x^2) = 2\log|x|$

and

2. $\ln(\cdot) := \log_e(\cdot)$ whereas $\log(\cdot) := \log_{10}(\cdot)$

Although for (2), popular sources such as wolfram define $\log(\cdot)$ as $\log_e(\cdot)$. The complex logarithm $\log(z)$ also assumes a base of $e$. Knowing the context and source is important.