Show that for any positive numbers $b$, $d$ and for any positive numbers $a$, $c$ that are n

Question

Show that for any positive numbers $b$, $d$ and for any positive numbers $a$, $c$ that are not equal to $1$, it holds that $\left( {{\log }_{a}}b \right)\left( {{\log }_{c}}d \right)=\left( {{\log }_{a}}d \right)\left( {{\log }_{c}}b \right)$.

Accepted Answer

Both sides equal $\dfrac{\ln b\,\ln d}{\ln a\,\ln c}$.

Exponential and Logarithmic Functions - Laws of Logarithms

Exponential and Logarithmic Functions - Laws of Logarithms

Solution:

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