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

Skilled2021-10-24Added 96 answers

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

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

Hence, the given expression as a single logarithm is \log_29