glamrockqueen7

2021-10-23

Use the change-of-base formula to write $\left({\mathrm{log}}_{25}\right)\left({\mathrm{log}}_{59}\right)$ as a single logarithm.

Caren

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 $\left({\mathrm{log}}_{25}\right)\left({\mathrm{log}}_{59}\right)$. The change-of-base formula is ${\mathrm{log}}_{a}b=\frac{\mathrm{ln}b}{\mathrm{ln}a}$ where ln represents natural logarithm. Use this formula to express given expression as a single logarithm.
$\left({\mathrm{log}}_{25}\right)\left({\mathrm{log}}_{59}\right)=\frac{\mathrm{ln}5}{\mathrm{ln}2}\cdot \frac{\mathrm{ln}9}{\mathrm{ln}5}$
$=\frac{\mathrm{ln}9}{\mathrm{ln}2}$
$={\mathrm{log}}_{29}$
Hence, the given expression as a single logarithm is \log_29

Do you have a similar question?