Question

How do you condense $2\log a+\log b-3\log c-\log d$?

Answer 1

Hint: The expression given to us is to be reduced in the simplest form. We will first use the property of the logarithm, that is, \[x\log a=\log {{a}^{x}}\]. The then obtained expression, which is, \[\log {{a}^{2}}+\log b-\log {{c}^{3}}-\log d\], is further simplified using the logarithmic formulae, which are, \[\log a+\log b=\log ab\] and \[\log a-\log b=\log \dfrac{a}{b}\]. Using these formulae wherever applicable, we will get the condensed form of the expression.

Complete step by step solution:
According to the given question, we have the expression which we have to condense. The expression has a logarithm function, so we will be using logarithm property to reduce the expression.
We will begin by writing the given expression first, we have,
$2\log a+\log b-3\log c-\log d$----(1)
We want to write each of the terms in the above expression in terms of log function, for that we will be using the logarithm property \[x\log a=\log {{a}^{x}}\], we get,
\[\Rightarrow \log {{a}^{2}}+\log b-\log {{c}^{3}}-\log d\]-----(2)
We know that, \[\log a+\log b=\log ab\].
So, we will now be considering the terms \[\log {{a}^{2}}\] and \[\log b\], and the use above formula, we get the expression as,
\[\Rightarrow (\log {{a}^{2}}+\log b)-\log {{c}^{3}}-\log d\]
\[\Rightarrow (\log ({{a}^{2}}.b))-\log {{c}^{3}}-\log d\]-----(3)
Now, we will consider the terms, \[\log {{c}^{3}}\]and \[\log d\], we get,
\[\Rightarrow (\log ({{a}^{2}}b))-(\log {{c}^{3}}+\log d)\]
Using the formula, \[\log a+\log b=\log ab\], we get,
\[\Rightarrow (\log ({{a}^{2}}b))-(\log ({{c}^{3}}.d))\]-----(4)
We have the expression which can be further reduced. We will now use one of the logarithmic equations, which is, \[\log a-\log b=\log \dfrac{a}{b}\].
Using this in equation (4), we get,
\[\Rightarrow \log ({{a}^{2}}b)-\log ({{c}^{3}}d)\]
\[\Rightarrow \log \left( \dfrac{{{a}^{2}}b}{{{c}^{3}}d} \right)\]
Therefore, the condensed form of the given equation is:
\[2\log a+\log b-3\log c-\log d=\log \left( \dfrac{{{a}^{2}}b}{{{c}^{3}}d} \right)\]

Note: The given expression had coefficients on logarithm function and we wanted the terms to be in terms of log alone, that is why we used the formula \[x\log a=\log {{a}^{x}}\] and hence we got the terms with log as shown in equation (2). Now, we could easily use the other properties or formula pertaining to logarithm function.