Key Concepts and Formulas

1. Logarithm Properties: $\log_b A = c \iff A = b^c$, $c \log_b a = \log_b a^c$, and if $\log_b A = \log_b B$, then $A=B$. The argument of a logarithm must be positive ($A > 0$).
2. Domain and Range: The domain is the set of allowed input values, and the range is the set of corresponding output values.
3. Geometric Progression (GP): A sequence where each term is found by multiplying the previous one by a constant common ratio $r$.
4. Sum of an Infinite GP: If $|r| < 1$, the sum of an infinite GP is $S_\infty = \frac{a}{1-r}$, where $a$ is the first term.

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Step-by-Step Solution

Step 1: Rewrite the logarithmic relation into an exponential form. The given relation is $\log_{\mathrm{e}} y = x \log_{\mathrm{e}}\left(\frac{2}{5}\right)$. We can use the logarithm property $c \log_b a = \log_b a^c$ to rewrite the right side of the equation. $\log_{\mathrm{e}} y = \log_{\mathrm{e}}\left(\left(\frac{2}{5}\right)^x\right)$ Now, using the property that if $\log_b A = \log_b B$, then $A=B$, we can equate the arguments of the logarithm. $y = \left(\frac{2}{5}\right)^x$ For the original logarithmic expression $\log_e y$ to be defined, we must have $y > 0$. Since $\left(\frac{2}{5}\right)^x$ is always positive for any real $x$, this condition is satisfied.

Step 2: Determine the elements of the range of R. The domain for $x$ is given as $S = \mathbf{N} \cup \{0\}$, which means $x$ can take values $0, 1, 2, 3, \ldots$. We substitute these values into the expression for $y$ to find the elements of the range. For $x=0$: $y = \left(\frac{2}{5}\right)^0 = 1$. For $x=1$: $y = \left(\frac{2}{5}\right)^1 = \frac{2}{5}$. For $x=2$: $y = \left(\frac{2}{5}\right)^2 = \frac{4}{25}$. For $x=3$: $y = \left(\frac{2}{5}\right)^3 = \frac{8}{125}$. The set of all elements in the range of $R$ is $\{1, \frac{2}{5}, \frac{4}{25}, \frac{8}{125}, \ldots\}$.

Step 3: Identify the sequence of range elements as an infinite geometric progression and calculate its sum. The sequence of elements in the range is $1, \frac{2}{5}, \frac{4}{25}, \frac{8}{125}, \ldots$. This is a geometric progression with: The first term, $a = 1$. The common ratio, $r = \frac{\text{second term}}{\text{first term}} = \frac{2/5}{1} = \frac{2}{5}$. We check if the sum of this infinite GP converges. The condition for convergence is $|r| < 1$. Here, $|r| = |\frac{2}{5}| = \frac{2}{5}$, which is less than 1. Therefore, the sum converges. The sum of an infinite geometric progression is given by the formula $S_\infty = \frac{a}{1-r}$. Substituting the values $a=1$ and $r=\frac{2}{5}$: $S_\infty = \frac{1}{1 - \frac{2}{5}} = \frac{1}{\frac{5}{5} - \frac{2}{5}} = \frac{1}{\frac{3}{5}} = \frac{5}{3}$ The sum of all the elements in the range of $R$ is $\frac{5}{3}$.

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Common Mistakes & Tips

• Logarithm Domain: Ensure the argument of any logarithm is strictly positive. In this case, $y>0$, which is satisfied by the derived expression.
• Base Case $x=0$: Remember that any non-zero number raised to the power of 0 is 1. This is crucial for identifying the first term of the GP correctly.
• Infinite GP Convergence: Always verify that $|r| < 1$ before applying the sum formula for an infinite geometric progression.

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Summary

The problem involves converting a logarithmic relation into an exponential form to express $y$ in terms of $x$. By considering the specified domain for $x$, we generated a sequence of $y$ values which formed an infinite geometric progression. The sum of this progression was calculated after confirming its convergence. The sum of all elements in the range of $R$ is $\frac{5}{3}$.

The final answer is $\boxed{\frac{5}{3}}$ which corresponds to option (D).