Question regarding logarithms

Question

Can you factor out the $m$ out of $\ln(c\cdot x^m)$ where $c$ is a constant?

Answer 1

• We use the fact that $\ln(ab) = \ln a + \ln b$,
• and the fact that $\ln(a^b) = b\ln a$

$$\ln(cx^m) = \ln(c) + \ln(x^m) = \ln(c) + m\ln(x)$$

Answer 2

If we have $y=ln(c.x^m)$, then $y=ln(c)+mln(x)$, so $\dfrac{y-ln(c)}{m}=ln(x)$ and so:

$$x=e^{\dfrac{y-ln(c)}{m}}$$

Answer 3

If $m\not = 0$, then we have $$ \ln\left(cx^m\right) = \ln c + \ln x^m = \ln c + m\ln x = m\left(\frac1m\ln c + \ln x\right)$$