Use the change-of-base formula to write (log25)(log59) as a single logarithm.

Step 1
For natural logarithms the base is e. This is the most commonly used base for logarithm.
Another common base for logarithms is 10. But a logarithm can have any positive real number as its base. And using the change-of-base formula the logarithm of any base can be expressed in terms of natural logarithms.
Step 2
The given expression to be simplified is ${\displaystyle \left({\log }_{25}\right)\left({\log }_{59}\right)}$. The change-of-base formula is ${\displaystyle {\log }_{a}b=\frac{\ln b}{\ln a}}$ where ln represents natural logarithm. Use this formula to express given expression as a single logarithm.
${\displaystyle \left({\log }_{25}\right)\left({\log }_{59}\right)=\frac{\ln 5}{\ln 2}\cdot \frac{\ln 9}{\ln 5}}$
${\displaystyle =\frac{\ln 9}{\ln 2}}$
${\displaystyle ={\log }_{29}}$
Hence, the given expression as a single logarithm is \log_29