How to Solve Log Equations with a Variable Base

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Homework Help Overview

The problem involves solving a logarithmic equation with a variable base, specifically Log [base x] 216 = 6, and finding the value of x.

Discussion Character

  • Exploratory, Assumption checking

Approaches and Questions Raised

  • Participants discuss converting the logarithmic equation into exponential form and express confusion about the next steps. There are questions about the necessity of using e or ln in this context, and some suggest taking the sixth root of 216 as a potential approach.

Discussion Status

The discussion is ongoing, with participants exploring different interpretations of the problem. Some guidance has been offered regarding the simplification of the equation, but there is no consensus on the method to achieve an exact answer.

Contextual Notes

There is mention of a requirement for an exact decimal answer, which some participants question, indicating a possible misunderstanding of the problem's expectations.

ilygurlie
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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!
 
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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.
 


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


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..
 


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


An "exact decimal answer"? There is none!
 

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