How to Solve Log Equations with a Variable Base

[ilygurlie]

Homework Statement

Log [with base x]216 =6
Find x

Homework Equations

The Attempt at a Solution

Ok, so I put the log equation into exponential form.
It now becomes:
x^6=216
I'm lost on what to do next.. I've tried taking the e and ln of both sides, but when I check it, it's not right. Do I just plug it into my calc? I thought there was some way to do this algebracally... please help!

 

[jacksonpeeble]

To the best of my knowledge, this is as simplified as it can be unless you want a decimal answer, in which case you'd just take the sixth root of 216 on your calculator.

What were you actually asked to do? Just find X? Or something else?

There is no reason to use e or ln in this equation. Those are only used in equations that start off with either e or a natural log (typically when working with continually compounded interest) - you don't just bring them in arbitrarily.

 

[Dick]

Take both sides of x^6=216 to the power of (1/6).

 

[ilygurlie]

Yeah, I'm really confused... My teacher wants an exact, decimal answer so I guess I'll just do it with my calc... but I really thought that was some way without it..

 

[Dick]

It's (216)^(1/6). You can simplify that if you factor 216=2^3*3^3.

 

[HallsofIvy]

An "exact decimal answer"? There is none!