(Note: I'm preparing for an Professional Engineering exam on topics I have not used in over 30 years. So questions might be overly basic. My apologies.)
1) If I take the natural log of one side of an equation, do I have to do so with the other side as well, to not change the equality?
2) If I take the natural log of one element of a multi-element equation, must I take it for all elements so as not to change the equality.
Example 1: For Equation: $\delta S = mC_p(T_2 – T_1)$, can I take the natural log of the right side only without changing the equality to get this equation?: $\delta S = (S_2 – S_1) = mC_pln(T_2/T_1)$.
Example 2: For Equation: $\delta S = mC_p(T_2 – T_1) – (P_2 – P_1)$, can I take the natural log of the far right element only without changing the equality to get this equation?: $\delta S = (S_2 – S_1) = mC_p(T_2 – T_1) – ln(P_2/P_1)$.
Edit 1: Another way of asking this question… In the equation "y = x*x – a + b" we can factor it to read "y = x^2 – a + b" where we change only the element immediately after the "=" sign (by squaring "x") without changing the equality.
Best Answer
I like to think of logarithms as exponents or taking the square root of equations. So for your parts if:
$$ a=b \implies \ln(a)=\ln(b) $$ also if
$$ a+b=c+d \implies \ln(a+b)=\ln(c+d) $$
there are also some other cool properties that might be useful for you
$$ \ln(a^b)=b\ln(a) $$ $$ \ln\left(\dfrac{a}{b}\right)=\ln(a)-\ln(b) $$ $$ \ln(ab)=\ln(a)+\ln(b) $$ $$ e^{\ln(a)}=a \implies b^{\log_b(a)}=a $$
These properties also hold true for any base of logarithm