Question

How do I find $y$ as a function of $x$ ? The constant $C$ is a positive number. $\ln (y - 3) = \ln 2{x^2} + \ln C$

Answer 1

Hint : To solve this type of question we should know about the properties of $\ln$ and the property and characteristic of $e$ .
Four basic properties of logs:
$\log \left( {\dfrac{x}{y}} \right) = \log x - \log y$
$\log (xy) = \log x + \log y$
$\log {x^n} = n\log x$
${\log _b}x = \dfrac{{{{\log }_a}x}}{{{{\log }_a}b}}$

Complete step by step solution:
Step 1: try to make a complex equation into a simpler one.
In this case, take R.H.S. and add.
$\ln 2{x^2} + \ln c = \ln \left( {2C{x^2}} \right)$
Step 2: take $e$ on both sides. We get,
$\ln (y - 3) = \ln (2C{x^2})$
$\Rightarrow {e^{\ln (y - 3)}} = {e^{\ln (2C{x^2})}}$
As we know, ${e^{\ln x}} = x$
So,
$\Rightarrow y - 3 = 2C{x^2}$
Step 3: by taking $y$ on one side and other on the other side. We get,
$y = 2C{x^2} + 3$
Hence, $y = 2C{x^2} + 3$ ,this is an equation of $y$ as a function of $x$ .
So, the correct answer is “$y = 2C{x^2} + 3$”.

Note : As $\log x$ means the base $10$ logarithm, It can also be written as ${\log _{10}}(x)$ . Logarithms can be defined for any positive base other than $1$ , not only $e$ . However, logarithms in other bases differ only by a constant multiplier from the natural logarithm, and can be defined in terms of the latter.