Is there a convention against having a fraction passed to a logarithmic function? The reason I'm asking is for my Calculus I class.
My teacher wants me to solve for x, and counts my answer as incomplete. Here's an example:
Solve for x:
e^x = 1/2
My answer: x = ln(1/2)
Teacher's answer: -ln(2)
I understand the law where ln(1/2) = ln(1) - ln(2) technically making my answer and my teacher's answer both correct.
And thus my question. Should my answer be accepted as correct, or is there a convention against having fractions as an argument to a logarithmic function?
Thanks!
$$\ln\left(\dfrac{a}{b}\right)=\ln a-\ln b$$
For your case $\begin{bmatrix}a \\ b\end{bmatrix}=\begin{bmatrix}1 \\2\end{bmatrix}$. Since $\ln 1=0$. We have:
$$\begin{aligned} x&=\ln(1/2)\\&=\ln1-\ln2\\&=-\ln2\end{aligned}\implies \ln(1/2)\equiv -\ln(2)$$
No there is no such convention. So yes your answer is equivalent to your instructor's. But here's a suggestion: it's always better to simplify the expression further.