Question

Condense \[2{\log _3}\left( x \right) + 5{\log _3}\left( y \right) - 4{\log _3}\left( z \right)\]?

Answer 1

Hint: The given question can be solved by using properties of logarithms such as \[\ln x + \ln y = \ln \left( {xy} \right)\], \[{\log _a}{x^n} = n{\log _a}x\] and \[\log x - \log y = \log \left( {\frac{x}{y}} \right)\] after applying the properties then solve the expression to get the required result.

Complete step by step answer:
A logarithm is an exponent which indicates to what power a base must be raised to produce a given number.
 \[y = {b^x}\] exponential form,
\[x = {\ln _b}y\] logarithmic function, where \[x\] is the logarithm of \[y\] to the base \[b\], and \[{\log _b}y\] is the power to which we have to raise \[b\]to get\[y\], we are expressing\[x\]in terms of \[y\].
Given logarithmic function is \[2{\log _3}\left( x \right) + 5{\log _3}\left( y \right) - 4{\log _3}\left( z \right)\],
Now applying the logarithmic identity \[\ln {x^n} = n\ln x\] for all the three terms, we get,
\[ \Rightarrow {\log _3}{x^3} + {\log _3}{y^5} - {\log _3}{z^4}\],
Now again applying the logarithmic identity \[\ln x + \ln y = \ln \left( {xy} \right)\] for the first two terms i.e., \[{\log _3}{x^3}\]and \[{\log _3}{y^5}\], we get,
\[ \Rightarrow {\log _3}\left( {{x^3} \cdot {y^5}} \right) - {\log _3}{z^4}\],
Now again applying the logarithmic identity \[\log x - \log y = \log \left( {\frac{x}{y}} \right)\], for the \[{\log _3}\left( {{x^3} \cdot {y^5}} \right)\] and the third remaining term i.e., \[{\log _3}{z^4}\], we get,
\[ \Rightarrow {\log _3}\left( {\frac{{{x^3} \cdot {y^5}}}{{{z^4}}}} \right)\],
So, the simplified form of the given function is equal to \[{\log _3}\left( {\frac{{{x^3} \cdot {y^5}}}{{{z^4}}}} \right)\].

\[\therefore \] The condensed form of the given function \[2{\log _3}\left( x \right) + 5{\log _3}\left( y \right) - 4{\log _3}\left( z \right)\] will be equal to \[{\log _3}\left( {\frac{{{x^3} \cdot {y^5}}}{{{z^4}}}} \right)\].

Note: A logarithm is a mathematical operation that determines how many times a certain number, called the base, is multiplied by itself to reach another number, we know that logarithm is the power to which a number must be raised in order to get some other number, and the base unit is the number being raised to a power. In these types of questions, we will make use of logarithmic properties and formulas, and some of the useful formulas are:
 \[{\log _a}xy = {\log _a}x + {\log _a}y\],
\[\log x - \log y = \log \left( {\frac{x}{y}} \right)\],
\[{\log _a}{x^n} = n{\log _a}x\],
\[{\log _a}b = \frac{{{{\log }_e}b}}{{{{\log }_e}a}}\],
\[{\log _{\frac{1}{a}}}b = - {\log _a}b\],
\[{\log _a}a = 1\],
\[{\log _{{a^x}}}b = \frac{1}{x}{\log _a}b\].