How do solve this exponential equation for the variable?: 36^x= 1/216 My teacher told me to use natural log: ln My teacher also told me to use log base: I guess that is e

Expert Answers
violy eNotes educator| Certified Educator

We can rewrite 36 as an exponential with base 6. 

Take note that 6*6 = 36. So, 36 = 6^2. 

We can also rewrite 216 as an exponential with base 6. 

Take note that 6*6*6 = 216. So, 216 = 6^3.

So, we will have:

`6^(2x) = 1/6^3`

We can use the property: 1/a^m = a^-m. 

`6^(2x) = 6^-3`

Take the natural logarithm of both sides.

`ln6^(2x) = ln6^-3`

Use the property: lna^b = blna.

`(2x)ln6 = -3ln6`

Divide both sides by ln6.

`2x = -3`

Divide both sides by 2.

`x = -3/2`

Therefore, the answer is x = -3/2.