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(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.

Glenn
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    Equality is a strong condition. Think of it intuitively that if $A=B$ then $A$ is exactly the same as $B$, so if you change only one side of the equation in all likelihood your equality will cease to hold. Thus if you want to keep an equality I recommend you always apply the same function to both sides. – K.Power Mar 16 '17 at 20:36

1 Answers1

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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

jameselmore
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wjmccann
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