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 express \(\log_5 8\) 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 write: \[ \log_5 8 = \frac{\log_{10} 8}{\log_{10} 5} \] ### Step 3: Expressing \(\log_{10} 8\) and \(\log_{10} 5\) Next, we need to express \(\log_{10} 8\) in terms of \(X\). We know that: \[ 8 = 2^3 \implies \log_{10} 8 = \log_{10} (2^3) = 3 \log_{10} 2 \] Now, we need to express \(\log_{10} 5\). We can use the fact that \(10 = 2 \times 5\): \[ \log_{10} 10 = \log_{10} (2 \times 5) = \log_{10} 2 + \log_{10} 5 \] Since \(\log_{10} 10 = 1\), we can rearrange this to find \(\log_{10} 5\): \[ \log_{10} 5 = 1 - \log_{10} 2 \] ### Step 4: Substituting Back Now we can substitute these expressions back into our equation for \(\log_5 8\): \[ \log_5 8 = \frac{3 \log_{10} 2}{1 - \log_{10} 2} \] ### Step 5: Relating \(\log_{10} 2\) to \(X\) From our original equation \(X = \log_8 10\), we can express \(\log_{10} 2\) in terms of \(X\). We know: \[ X = \frac{\log_{10} 10}{\log_{10} 8} = \frac{1}{3 \log_{10} 2} \] Rearranging gives: \[ \log_{10} 2 = \frac{1}{3X} \] ### Step 6: Substitute \(\log_{10} 2\) into \(\log_5 8\) Now we substitute \(\log_{10} 2\) back into our expression for \(\log_5 8\): \[ \log_5 8 = \frac{3 \left(\frac{1}{3X}\right)}{1 - \left(\frac{1}{3X}\right)} = \frac{1}{X}{1 - \frac{1}{3X}} = \frac{1}{X} \cdot \frac{3X - 1}{3X} = \frac{3X - 1}{3X^2} \] ### Step 7: Final Expression Now we can simplify this expression. We want to express \(\log_5 8\) in terms of \(X\): \[ \log_5 8 = \frac{3X - 1}{3X} \] This can be rewritten as: \[ \log_5 8 = X - \frac{1}{3} \] ### Conclusion Thus, we find that: \[ \log_5 8 = X - \frac{1}{3} \] This corresponds to option C. ### Summary of the Solution - We used the change of base formula to express \(\log_5 8\). - We expressed \(\log_{10} 8\) and \(\log_{10} 5\) in terms of \(\log_{10} 2\). - We derived \(\log_{10} 2\) from the given \(X\). - Finally, we simplified to find \(\log_5 8 = X - \frac{1}{3}\). ### Revision Points - Remember the change of base formula for logarithms. - Be familiar with expressing logarithms of powers (e.g., \(\log_{10} (2^3)\)). - Understand how to manipulate logarithmic identities to express one logarithm in terms of another. - Practice deriving relationships between different logarithmic bases.