Question

If $\ln y + \ln x = \ln C$, how do you find an expression for y?

Answer 1

Hint: According to the question given in the question we have to determine an expression for y if $\ln y + \ln x = \ln C$ as asked in the question. So, first of all we have to solve the given logarithmic expression which is $\ln y + \ln x = \ln C$ for which we have to isolate the term y so we have to move all the terms containing y in any one of the sides.
Now, to make the expression in which y is isolated we have to subtract $\ln x$ in the both sides of the expression given $\ln y + \ln x = \ln C$.
Now, to solve the expression obtained we have to use the property of logarithm which is when the two terms are written in form of subtraction then we can also write the terms in the form of their division we can also understand with the help of formula which is as mentioned below:

Formula used: $ \Rightarrow \ln a - \ln b = \ln \left( {\dfrac{a}{b}} \right)..........(A)$
As from the formula (A) above, we can see that the subtraction of two logarithm terms can be represented as their division.
Now, to solve the expression in terms of y we have to use the formula which is as mentioned below:
$ \Rightarrow $if $a = \ln b$ then ${e^a} = b$………………….(B)
Now, we have to simplify the expression in the form of power of e using the rule which is as explained below:
$ \Rightarrow {e^{\ln (a)}} = a$f or any a………………..(C)

Complete step-by-step solution:
Step 1: First of all we have to solve the given logarithmic expression which is $\ln y + \ln x = \ln C$ for which we have to isolate the term y so we have to move all the terms containing y in any one of the sides. Which is as explained in the solution hint. Hence,
Step 2: Now, to make the expression in which y is isolated we have to subtract $\ln x$ in the both sides of the expression given $\ln y + \ln x = \ln C$. Hence,
$
   \Rightarrow \ln y + \ln x - \ln x = \ln C - \ln x \\
   \Rightarrow \ln y = \ln C - \ln x............(1)
 $
Step 3: Now, to solve the expression obtained in the solution step 2 we have to use the property of logarithm which is when the two terms are written in form of subtraction then we can also write the terms in the form of their division we can also understand with the help of formula (A) which is as mentioned in the solution hint. Hence,
$ \Rightarrow \ln y = \ln \dfrac{C}{x}...............(2)$
Step 4: Now, to solve the expression (2) which is as obtained in the solution step 3, in terms of y we have to use the formula (B) which is as mentioned in the solution hint. Hence,
$ \Rightarrow {e^{\ln \left( {\dfrac{c}{x}} \right)}} = y.................(3)$
Step 5: Now, we $ \Rightarrow \dfrac{c}{x} = y$ have to simplify the expression in the form of power of e using the rule (C) which is as explained in the solution hint.

Hence, with the help of the formulas (A), (B), and (C) we have obtained the expression in the terms of y which is $ \Rightarrow \dfrac{c}{x} = y$.

Note: If there are two or more than two logarithm terms are written in there form of sum or addition as $\ln x + \ln y + \ln z$ then we can represent them in the form of their multiplication as $\ln (xyz)$ and if the same terms are written in the form of their subtraction as $\ln x - \ln y$ then we can represent them in the form of their division as $\ln \left( {\dfrac{x}{y}} \right)$.
If it is asked in the question that the required expression should be in the terms of y then we have to take all the terms containing y in the same side of the given logarithm expression.