Question

How do you condense \[2\ln x + 3\ln y - 6\ln z\] ?

Answer 1

Hint: The given function is the logarithm function it can be defined as logarithmic functions are the inverses of exponential functions. By using some different Basic Properties of logarithmic, on step by step simplification, we get the required solution in the function as a single logarithm function.

Complete step by step answer:
The function from positive real numbers to real numbers to real numbers is defined as \[{\log _b}:{R^ + } \to R \Rightarrow {\log _b}\left( x \right) = y\], if \[{b^y} = x\], is called logarithmic function or the logarithm function is the inverse form of exponential function.
There are some basic logarithms properties
1. product rule :- \[\log \left( {mn} \right) = \log m + \log n\]
2. Quotient rule :- \[\log \left( {\dfrac{m}{n}} \right) = \log m - \log n\]
3. Power rule :- \[\log \left( {{m^n}} \right) = n.\log m\]
Now, Consider the logarithm function
 \[ \Rightarrow \,\,2\ln x + 3\ln y - 6\ln z\]------ (1)
Apply power rule for all the three terms, then equation (1) becomes
\[ \Rightarrow \,\,\ln {x^2} + \ln {y^3} - \ln {z^6}\]------- (2)
Now, using the product rule for the first two terms in equation (2), then
\[ \Rightarrow \,\,\ln \left( {{x^2} \cdot {y^3}} \right) - \ln {z^6}\]---------(3)
Using quotient rule we can write the equation (3) as
\[ \Rightarrow \,\,\ln \left( {\dfrac{{{x^2} \cdot {y^3}}}{{{z^6}}}} \right)\]
Hence, the required condensed form of the given function \[2\ln x + 3\ln y - 6\ln z\] is \[\ln \left( {\dfrac{{{x^2} \cdot {y^3}}}{{{z^6}}}} \right)\].
i.e., we condense three logarithmic functions to single logarithm function. We can also verify this by reverse process by applying the logarithm property. It’s a vice versa process.

Note: If the function contains the log term, then the function is known as logarithmic function. We have two types of logarithms namely common logarithm and natural logarithm. Since it is involving the arithmetic operations, we have a standard logarithmic property for the arithmetic operations. By using the properties, we can solve these types of questions.