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The concept of natural logarithms is fundamental when dealing with exponential equations. A natural logarithm, denoted as \( \ln(x) \), is the logarithm to the base \( e \), where \( e \approx 2.71828 \). This type of logarithm is often used in mathematics and science because of its natural occurrence in growth processes and continuous compounding, among other applications.

Natural logarithms provide an efficient way to simplify computations involving exponentials. In our problem, \( \ln(10^{1-x}) = \ln(6^{x}) \), taking the natural log of both sides allows us to bring the exponent out in front as a multiplier, thanks to the power rule. This transformation is key in solving for the variable \( x \) because it changes the equation from an exponential form to an algebraic form.

Using \( \ln \) helps to deal with complex exponential equations like the one in our exercise by turning multiplicative relationships into additive ones. This characteristic drastically simplifies solving for unknowns and is a powerful tool in algebra.

Applying the power rule for logarithms is crucial in simplifying equations that have variables in the exponent. The power rule states that \( \ln(a^b) = b\ln(a) \). It allows us to move the exponent in front of the logarithm as a coefficient.

In our equation \( 10^{1-x} = 6^{x} \), applying the power rule transforms it into \( (1-x)\ln(10) = x\ln(6) \). This is achieved by expressing each exponential's exponent as a product with the natural log of its base.

This manipulation leverages the additive properties of logarithms, making it easier to rearrange and isolate the variable \( x \). Hence, the power rule is an essential step in switching from exponential notation to an algebraic equation, simplifying what would otherwise be a challenging equation to solve.

Solving the given equation involves several key steps. Let's recap the process to fix the steps in our understanding.

• **Step 1**: We started by taking the natural log of both sides: \( \ln(10^{1-x}) = \ln(6^{x}) \). This step reduces the complexity by transforming exponents into coefficients.


• **Step 2**: We applied the power rule to express the equation in linear terms: \( (1-x)\ln(10) = x\ln(6) \).


• **Step 3**: Distribute the log terms to each term inside the equation: \( \ln(10) - x\ln(10) = x\ln(6) \). It is important to expand carefully to prevent arithmetic mistakes.


• **Step 4**: Rearrange to collect all terms with \( x \) to one side: \( \ln(10) = x(\ln(6) + \ln(10)) \). Notice how factoring out \( x \) from a common factor simplifies rearranging.


• **Step 5**: Solve for \( x \) by isolating it: \( x = \frac{\ln(10)}{\ln(6) + \ln(10)} \). This step refines the algebraic manipulation to solve for \( x \).


• **Step 6**: Calculate the numerical value using a calculator to get \( x = 0.3865 \) to four decimal places.

The methodical approach of handling each step ensures that solving exponential equations becomes manageable. Following this structured method not only helps in solving the problem at hand but also aids in understanding and tackling similar problems in the future.