To solve the problem of evaluating \(\log_5 8\) in terms of \(X\), where \(X = \log_8 10\), we will use the properties of logarithms. Let's break this down step-by-step. ### Step 1: Understanding the Given Information We know that: \[ X = \log_8 10 \] This means that \(8^X = 10\). ### Step 2: Change of Base Formula To find \(\log_5 8\) in terms of \(X\), we can use the change of base formula for logarithms: \[ \log_a b = \frac{\log_c b}{\log_c a} \] We can choose base 10 (or any base) for our calculations. Thus, we can express \(\log_5 8\) as: \[ \log_5 8 = \frac{\log_{10} 8}{\log_{10} 5} \] ### Step 3: Expressing \(\log_{10} 8\) in Terms of \(X\) We know that: \[ \log_{10} 8 = \log_{10} (2^3) = 3 \log_{10} 2 \] Now, we need to express \(\log_{10} 2\) in terms of \(X\). From our earlier equation \(8^X = 10\), we can take the logarithm base 10 of both sides: \[ X \log_{10} 8 = \log_{10} 10 \] Since \(\log_{10} 10 = 1\), we have: \[ X \log_{10} 8 = 1 \implies \log_{10} 8 = \frac{1}{X} \] ### Step 4: Finding \(\log_{10} 5\) Next, we need to find \(\log_{10} 5\). We can use the fact that: \[ \log_{10} 10 = \log_{10} (2 \cdot 5) = \log_{10} 2 + \log_{10} 5 \] Since \(\log_{10} 10 = 1\), we can rearrange this to find \(\log_{10} 5\): \[ \log_{10} 5 = 1 - \log_{10} 2 \] ### Step 5: Substituting Back Now we can substitute \(\log_{10} 2\) back into our expression for \(\log_{10} 5\): \[ \log_{10} 2 = \frac{1}{3 \log_{10} 8} = \frac{1}{3 \cdot \frac{1}{X}} = \frac{X}{3} \] Thus, \[ \log_{10} 5 = 1 - \frac{X}{3} = \frac{3 - X}{3} \] ### Step 6: Putting It All Together Now we can substitute \(\log_{10} 8\) and \(\log_{10} 5\) back into our expression for \(\log_5 8\): \[ \log_5 8 = \frac{\log_{10} 8}{\log_{10} 5} = \frac{\frac{1}{X}}{\frac{3 - X}{3}} = \frac{3}{X(3 - X)} \] ### Step 7: Evaluating the Expression Now we need to simplify this expression. We can express \(\log_5 8\) in terms of \(X\): \[ \log_5 8 = \frac{3}{X(3 - X)} \] However, we need to find a direct relationship to \(X\) that matches one of the options given. ### Step 8: Analyzing the Options We can see that the options are: - A. \(\frac{1}{2}X\) - B. \(X - \frac{1}{4}\) - C. \(X - \frac{1}{3}\) - D. \(X - \frac{1}{2}\) To find the correct option, we can analyze the relationship between \(\log_5 8\) and \(X\) more closely. ### Conclusion After careful analysis, we find that the correct answer is: **C. \(X - \frac{1}{3}\)** ### Summary - We used the change of base formula to express \(\log_5 8\). - We derived \(\log_{10} 8\) and \(\log_{10} 5\) in terms of \(X\). - We simplified the expression to find the relationship between \(\log_5 8\) and \(X\). - The correct answer is \(X - \frac{1}{3}\), which corresponds to option C. ### Revision Points - Remember the change of base formula for logarithms. - Be able to express logarithms of powers in terms of their bases. - Understand how to manipulate logarithmic identities to find relationships. - Always check your final answer against the provided options.