Solve log(base4)X + log(base4) (X+6) <2

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SUMMARY

The discussion focuses on solving the inequality log4X + log4(X+6) < 2. The key steps involve applying the logarithmic property loga(b*c) = logab + logac, which simplifies the inequality to log4(X(X+6)) < log4(16). This leads to the quadratic inequality X(X+6) < 16, ultimately yielding solutions X = -8 and X = 2. The participants emphasize the importance of determining valid values for X before solving the inequality.

PREREQUISITES
  • Understanding of logarithmic properties, specifically loga(b*c) = logab + logac.
  • Familiarity with solving quadratic inequalities.
  • Knowledge of the relationship between logarithmic and exponential forms, such as logab = c <==> b = ac.
  • Basic algebra skills for manipulating inequalities.
NEXT STEPS
  • Study the properties of logarithms in depth, focusing on loga(b*c) and its applications.
  • Practice solving quadratic inequalities to strengthen problem-solving skills.
  • Explore the relationship between logarithmic and exponential functions, particularly in solving equations.
  • Review examples of logarithmic inequalities and their graphical interpretations.
USEFUL FOR

Students studying algebra, particularly those tackling logarithmic inequalities, as well as educators seeking to enhance their teaching methods in logarithmic concepts.

VVVS
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Homework Statement


Solve log(base4)X + log(base4) (X+6) <2


Homework Equations


log(base4)X + log(base4) (X+6) =2 comes out to x=-8 and X=2


The Attempt at a Solution


I can't seem to figure out what steps I should even take to answer this inequality.
 
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this property should be useful:
log_a (b*c) = log_a b + log_a c
applying this to your inequality and changing 2 into log_4 16 you'll get something that can be easily transferred into inequality containing quadratic function, which you'll surely solve on your own
 


VVVS said:

Homework Statement


Solve log(base4)X + log(base4) (X+6) <2


Homework Equations


log(base4)X + log(base4) (X+6) =2 comes out to x=-8 and X=2
One relevant equation here would be logaA + logaB = logaAB.
Another is logab = c <==> b = ac.
The equation you have here is part of your work, not really a relevant equation.
VVVS said:

The Attempt at a Solution


I can't seem to figure out what steps I should even take to answer this inequality.

First off, you should determine which values of x are going to be allowed as potential solutions. Then, see if the equations I added might be of some help to you.
 

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