Steps on how to simplify log5/log125 to 1/3

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SUMMARY

The discussion focuses on simplifying the expression log5/log125 to 1/3 using logarithmic properties. Participants clarify that the base of the logarithm does not affect the simplification as long as it remains consistent. The key steps involve recognizing that 125 equals 5^3, allowing the transformation of log5/log125 into log5/(3*log5). This simplification leads to the cancellation of log5, resulting in the final answer of 1/3.

PREREQUISITES
  • Understanding of logarithmic properties, specifically the change of base formula.
  • Familiarity with exponentiation, particularly the relationship between bases and their powers.
  • Basic algebra skills for manipulating fractions and expressions.
  • Knowledge of logarithmic identities, such as log(a^b) = b*log(a).
NEXT STEPS
  • Study the properties of logarithms, including the product, quotient, and power rules.
  • Learn about the change of base formula for logarithms and its applications.
  • Explore advanced logarithmic equations and their simplifications.
  • Practice solving logarithmic expressions using different bases and properties.
USEFUL FOR

This discussion is beneficial for students, educators, and anyone looking to enhance their understanding of logarithmic simplifications and properties in mathematics.

Gughanath
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could someone show me the steps on how to simplify log5/log125 to 1/3. I can't do it
 
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Which base?
 
danne89 said:
Which base?
10, sorry for not mentioning that.
 
It doesn't matter the base,as long it is the same...:wink:

Daniel.
 
yes, so how could I simplify it?
 
125=5^{3} and use one definitory property of the logarithm...

Daniel.
 
hmmm...that comes to log5 to the power -2?
 
Why don't just: log5 125 = 3 and log5 5 = 1 and then just replace log 5 / log 125 = 1 / 3 ??
 
No,remember that in general:
\frac{\log a}{\log b}\neq \log\frac{a}{b}

So pay attention to what u do.
Daniel.
 
  • #10
log(125) = log(53) = 3log(5)
 
  • #11
oh right. My bad. so log5/log125 = log5/log5^3 = log5/3log5, the log5's cancel, leaving 1/3. Thanx
 
  • #12
And then cancel out the log(5)s, leaving 1/3?
 

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