Question:

Step 1: Apply the Change of Base Formula We begin by converting all logarithmic terms in the given equation to a common base, such as the natural logarithm (base $e$). The change of base formula states that $\log_b a = \frac{\log_c a}{\log_c b}$. Applying this formula: $$ \log_k x = \frac{\ln x}{\ln k} $$ $$ \log_5 k = \frac{\ln k}{\ln 5} $$ $$ \log_x 5 = \frac{\ln 5}{\ln x} $$ Step 2: Substitute the Converted Logarithms into the Original Equation Substitute these expressions back into the given equation $\log_k x \cdot \log_5 k = \log_x 5$: $$ \left(\frac{\ln x}{\ln k}\right) \cdot \left(\frac{\ln k}{\ln 5}\right) = \frac{\ln 5}{\ln x} $$ Step 3: Simplify the Equation Observe that the $\ln k$ terms in the product on the left-hand side cancel each other out. $$ \frac{\ln x}{\ln 5} = \frac{\ln 5}{\ln x} $$ Step 4: Cross-Multiply to Isolate $\ln x$ To eliminate the denominators, we cross-multiply the terms. $$ (\ln x)^2 = (\ln 5)^2 $$ Step 5: Solve for $\ln x$ Take the square root of both sides of the equation. Remember to consider both positive and negative roots. $$ \ln x = \pm \ln 5 $$ Step 6: Determine the Possible Values of $x$ We have two cases to consider based on the $\pm$ sign. Case 1: $\ln x = \ln 5$ If $\ln x = \ln 5$, then by the property of logarithms that if $\ln a = \ln b$, then $a = b$: $$ x = 5 $$ Case 2: $\ln x = -\ln 5$ Using the logarithm property $-\ln a = \ln(a^{-1}) = \ln\left(\frac{1}{a}\right)$: $$ \ln x = \ln\left(\frac{1}{5}\right) $$ Therefore: $$ x = \frac{1}{5} $$ Step 7: Verify the Validity of the Solutions We must ensure that the obtained values of $x$ satisfy the domain restrictions for logarithms, which require the base and argument to be positive and the base not equal to 1. The problem states $k \ne 1, k > 0$. For $x$, we must have $x > 0$ and $x \ne 1$. For $x = 5$: $5 > 0$ and $5 \ne 1$. This solution is valid. For $x = \frac{1}{5}$: $\frac{1}{5} > 0$ and $\frac{1}{5} \ne 1$. This solution is also valid. Step 8: Calculate the Sum of All Values of $x$ The sum of all valid values of $x$ is: $$ \text{Sum} = 5 + \frac{1}{5} $$ To add these fractions, find a common denominator: $$ \text{Sum} = \frac{25}{5} + \frac{1}{5} = \frac{25+1}{5} = \frac{26}{5} $$ The sum of all values of $x$ is $\frac{26}{5}$. The final answer is $\boxed{\frac{26}{5}}$.