 Let's solve the equation one-third to power x minus one equals 81 So this is a power equation. So we can hit both sides with a log And we can try to simplify over on the right hand side of the log of 81 And that's a number. I have no idea what that is. So I'll leave it alone Over on the left hand side I have one-third raised to power x minus one and one of our rules of logs is that if I'm going to take the log of a Power the exponent can be brought out front as a factor and So this exponent x minus one can be brought out front as a factor and my left hand side will simplify a Little analysis goes a long way over on the left hand side I have a product x minus one times log of one-third. So I can simplify a product by dividing And that gives me x minus one equals log 81 over log one-third I want to solve for x. So at this point I need to add one to both sides And I get my solution While this is a perfectly valid answer. We can simplify it if we make the following observations First 81 is itself a power. It's three to power four One-third is also a power of three. It's three to power negative one and this means we can do some Simplification if x is one plus log 81 over log one-third Equals means replaceable. So every place I see 81. I can replace it with three to the fourth Every place I see one-third I can replace it with three to power negative one but now I have powers and our Theorem says that if I want to take the log of a power the exponent can be brought out front as a multiplier So this log of three to power four that becomes four times the log of three And the log of three to power negative one becomes negative one times log three But wait, there's more in our fraction here both numerator and denominator have a common factor of log three And so we'll remove that common factor Do a little arithmetic And again while this is a perfectly good answer This is a perfectly good solution a better form of this perfectly good solution. It's just negative three