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Show that \[{{\log }_{b}}a{{\log }_{c}}b{{\log }_{a}}c=1.\]

Last updated date: 19th Jul 2024
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Hint: Use the base change property of logarithm to make each log with the same base . Change the base of each expression to either 10 or e.

Complete step-by-step answer:

We have the expression\[\left( {{\log }_{b}}a \right)\left( {{\log }_{c}}b \right)\left( {{\log }_{a}}c \right)\].
For a typical calculation, we can change logarithms to base 10 or e.
Let us use the base as 10 here.
\[\therefore {{\log }_{b}}a=\dfrac{{{\log }_{10}}a}{{{\log }_{10}}b}=\dfrac{\log a}{\log b}\]
\[{{\log }_{10}}a\] can be written as \[\log a.\]
Similarly, \[\begin{align}
  & {{\log }_{c}}b=\dfrac{{{\log }_{10}}b}{{{\log }_{10}}c}=\dfrac{logb}{\log c} \\
 & {{\log }_{a}}c=\dfrac{{{\log }_{10}}c}{{{\log }_{10}}a}=\dfrac{\log c}{\log a} \\
Substituting these values in the expression, we get,
\[\therefore \left( {{\log }_{b}}a \right)\left( {{\log }_{c}}b \right)\left( {{\log }_{a}}c \right)=\dfrac{\log a}{\log b}\times \dfrac{\log b}{\log c}\times \dfrac{\log c}{\log a}.\]
Now let us cancel out \[\log a,\log b,\log c\] from the numerator and denominator we get the answer as 1.
\[\left( {{\log }_{b}}a \right)\left( {{\log }_{c}}b \right)\left( {{\log }_{a}}c \right)=1.\]
Hence we have proved that \[\left( {{\log }_{b}}a \right)\left( {{\log }_{c}}b \right)\left( {{\log }_{a}}c \right)=1.\]

Note: We solved the problem using base k as 10. We know that the base can also be taken as k = e.
Then \[{{\log }_{b}}a=\dfrac{{{\log }_{e}}a}{{{\log }_{e}}b};{{\log }_{c}}b=\dfrac{{{\log }_{e}}b}{{{\log }_{e}}c};{{\log }_{a}}c=\dfrac{{{\log }_{e}}c}{{{\log }_{e}}a}.\]
By multiplying these, we still get the value as 1.
\[\therefore \left( {{\log }_{b}}a \right)\left( {{\log }_{c}}b \right)\left( {{\log }_{a}}c \right)=\dfrac{{{\log }_{e}}a}{{{\log }_{e}}b}\times \dfrac{{{\log }_{e}}b}{{{\log }_{e}}c}\times \dfrac{{{\log }_{e}}c}{{{\log }_{e}}a}=1.\]