When are you supposed to take the natural log of both sides of an equation?

[Jurrasic]

It seems this is done randomly, yet it must not be random, there must be some logic to it, but when do you know to do it? What is like a clue or the rules that indicate you can/should take the natural log of both sides of an equation?

 

[iRaid]

You use "e" to cancel out a natural log of an equation.

 

[Mark44]

It seems this is done randomly, yet it must not be random, there must be some logic to it, but when do you know to do it? What is like a clue or the rules that indicate you can/should take the natural log of both sides of an equation?

If you have an equation that involves exponents, that's a natural for taking the log of both sides.

For example, if e^{x + 2} = 3, we can take the natural log of both sides to get
ln(e^{x + 2}) = ln(3)
x + 2 = ln(3)
x = -2 + ln(3) ≈ -1.307

 

[symbolipoint]

It seems this is done randomly, yet it must not be random, there must be some logic to it, but when do you know to do it? What is like a clue or the rules that indicate you can/should take the natural log of both sides of an equation?

The key to understand is you use an inverse operation or a sequence of inverse operations to reveal a formula for an unknown but sought variable. An exponential function is the inverse of a logarithmic function. You would do something similar if you wanted to clear for an additive inverse or a multiplicative inverse.

Mark44's example to illustrate is a good one.

Note that log_b(b^x)=x