Proving 1/3 < log_{34} 5 < 1/2

In summary, The inequality "1/3 < log_{34} 5 < 1/2" means that the value of log_{34} 5 is between 1/3 and 1/2. It can be proven using algebraic manipulation and properties of logarithms. Proving this inequality is important for understanding the relationship between logarithms and fractions and has real-life applications. This inequality can also be generalized to other logarithmic expressions with a base greater than 1.
  • #1
ubergewehr273
142
5

Homework Statement


Prove that ##{1/3} < log_{34} 5 < {1/ 2}##

Homework Equations


##log_b a = {1/ log_a b}##
##logmn = logm + logn##

The Attempt at a Solution


##log_{34} 5 = {1/ log_5 34}##
##= 1/(log_5 17 + log_5 2)##
##=1/(1 + log_5 3 + log_5 2 + something)##
##=1/(1 + log_5 6 + something)##
##=1/(2+something~else)##

something else ##<1##

Hence it is greater than 1/3 and smaller than 1/2.
But I think mathematically speaking, that ##something## isn't supposed to be there.
 
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  • #2
A better way to do this is to recast the inequality as
[tex]3 > \log_{5}34 > 2[/tex]
and rewrite ##3## and ##2## in terms of base-5 logarithms as well.
 
  • Like
Likes RUber
  • #3
Thanks. Made it much easier than my solution.
 

FAQ: Proving 1/3 < log_{34} 5 < 1/2

1. What does the inequality "1/3 < log_{34} 5 < 1/2" mean?

The inequality "1/3 < log_{34} 5 < 1/2" means that the value of log_{34} 5 is between 1/3 and 1/2. In other words, log_{34} 5 is greater than 1/3 and less than 1/2.

2. How is this inequality proven?

This inequality can be proven using algebraic manipulation and properties of logarithms. First, we can rewrite the expression as log_{34} 5 = log_{34}(34^{1/3}) = 1/3. Then, we can use the property log_b (x^a) = a log_b (x) to rewrite log_{34} 5 as log_{34}(34^{1/3}) = 1/3 log_{34}(34) = 1/3. Finally, we can use the fact that log_b (b) = 1 to conclude that 1/3 < log_{34} 5 < 1/2.

3. Why is proving this inequality important?

Proving this inequality is important because it helps us understand the relationship between logarithms and fractions. It also has applications in various fields such as mathematics, science, and engineering.

4. Can this inequality be generalized to other logarithmic expressions?

Yes, this inequality can be generalized to other logarithmic expressions. As long as the base of the logarithm is greater than 1, the inequality will hold true. However, the specific values of 1/3 and 1/2 may change depending on the base and the expression.

5. Are there any real-life examples of this inequality?

Yes, there are many real-life examples of this inequality. One example is when calculating interest rates using compound interest, where the logarithmic expression represents the number of compounding periods. Another example is in the field of acoustics, where the decibel scale is based on logarithmic expressions and follows a similar inequality pattern.

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