How to prove this logarithmic inequality?

In summary, the conversation discusses how to prove the inequality $\displaystyle \left(\log_{24}(48) \right)^2+\displaystyle \left(\log_{12}(54) \right)^2 >4$. The approach involves changing the bases to base-10 log and using the properties of logarithms to simplify the expression. Ultimately, it is shown that the inequality holds true.
  • #1
anemone
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MHB
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Hi all, I've been having a hard time trying to solve the following inequality:

Prove that $\displaystyle \left(\log_{24}(48) \right)^2+\displaystyle \left(\log_{12}(54) \right)^2 >4$

I've tried to change the bases to base-10 log and relating all the figures (12, 24, 48, and 54) in terms of 2 and 3 but only to make the problem to be more confounded.

Could I get some hints on how to tackle this problem?

Any help would be deeply appreciated.

Thanks!

P.S. This question was originally asked here (http://www.mymathforum.com/viewtopic.php?f=13&t=27644&p=110515&hilit=noki#p110515) at MMF.
 
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  • #2
anemone said:
Hi all, I've been having a hard time trying to solve the following inequality:

Prove that $\displaystyle \left(\log_{24}(48) \right)^2+\displaystyle \left(\log_{12}(54) \right)^2 >4$

I've tried to change the bases to base-10 log and relating all the figures (12, 24, 48, and 54) in terms of 2 and 3 but only to make the problem to be more confounded.

Could I get some hints on how to tackle this problem?

Any help would be deeply appreciated.

Thanks!

P.S. This question was originally asked here (http://www.mymathforum.com/viewtopic.php?f=13&t=27644&p=110515&hilit=noki#p110515) at MMF.

\(\log_{24}(48)=1+\log_{24}(2)\)

But \(2^5 \gt 24\) so \(\log_{24}(2) \gt 1/5\)

Also: \(\log_{12}(54)=1+\log_{12}(4.5)\), and \(4.5^5>12^3\) so \(\log_{12}(4.5)>3/5\)

Hence:
\[ (\log_{24}(48))^2 + (\log_{12}(54))^2 \gt 1.2^2+1.6^2 =4 \]

CB
 
  • #3
Hi CB, a big thank for your help in making it so straightforward and simple for me!

Thanks.
 

Related to How to prove this logarithmic inequality?

1. How do I know which property or rule to use to prove a logarithmic inequality?

To prove a logarithmic inequality, you will need to use the properties and rules of logarithms, such as the product rule, quotient rule, and power rule. You will also need to use algebraic manipulation to simplify the expression and isolate the logarithmic term. The key is to carefully analyze the given inequality and choose the appropriate property or rule to apply.

2. Can I use a calculator to prove a logarithmic inequality?

While a calculator can help you evaluate logarithmic expressions, it is not necessary to use one to prove a logarithmic inequality. In fact, it is recommended to do the calculations by hand to better understand the steps involved in the proof.

3. Is it necessary to prove all logarithmic properties when proving an inequality?

No, it is not necessary to prove all logarithmic properties in order to prove an inequality. Depending on the given inequality, you may only need to use one or two properties to prove it. However, it is important to have a thorough understanding of all logarithmic properties in order to choose the correct ones for each proof.

4. How can I check my answer when proving a logarithmic inequality?

One way to check your answer is by plugging in values for the variables in the original inequality and the result of your proof. If the values satisfy the original inequality but not the result, then your proof is incorrect. You can also verify your proof by working backwards and substituting the proven expression into the original inequality.

5. Are there any common mistakes to watch out for when proving logarithmic inequalities?

Yes, some common mistakes to watch out for include forgetting to apply logarithmic properties, making algebraic errors, and not simplifying the expression enough. It is important to carefully check each step of the proof to ensure accuracy and avoid any mistakes.

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