How does $\ln \frac{1}{2}$ become $-\ln2$?

Question

How does $\ln \frac{1}{2}$ become $-\ln2$

I just saw a video that said you can transfer the two from the denominator to the numerator, but how does that make the 2 negative? Wouldn't that just make the exponent negative?

For example $\frac{1}{2}$= $2^{-1}$

or am I missing something?

Answer 1

Observe that $\ln(\frac{1}{2})=\ln(2^{-1})=-\ln(2)$. Since $2^{-1}=\frac{1}{2}$ and using the log rules for powers.

Answer 2

$$\ln\frac{1}{2}=\ln1-\ln2=0-\ln2=-\ln2$$

Answer 3

There is a rule that $\log_{b} a^n=n \log_{b} a$. Since $\log_{b} a^n=\log_{b}(a \times a \times ... \times a)=\log_{b} a+\log_b a+...+\log_{b} a=n \log_b a$