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.