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Natural logarithms are logarithms that use the number 'e' as the base, where 'e' is an irrational constant approximately equal to 2.71828. They are denoted by \( \ln(x) \), meaning the logarithm of \( x \) with base \( e \). Natural logarithms are important in calculus and various fields due to their close relation to exponential growth and decay processes.

When solving exponential equations like \( e^{-0.08k} = 10 \), taking the natural logarithm of both sides helps to simplify the equation. This transformation leverages the property that \( \ln(e^x) = x \), enabling us to isolate variables and solve for unknowns.

In our solution, we apply this by taking \( \ln(e^{-0.08k}) \) and \( \ln(10) \), effectively removing the exponential factor on one side of the equation.

An exact solution gives the precise answer to an equation without any approximations. It is typically represented in a mathematical form that fully satisfies the equation's conditions.

For the equation \( e^{-0.08k} = 10 \), we find the exact solution by isolating \( k \). After applying natural logarithms, the equation simplifies to \(-0.08k = \ln(10)\).

By solving for \( k \), we divide both sides by \(-0.08\), yielding the exact solution:

• \( k = \dfrac{\ln(10)}{-0.08} \)

This form maintains all the precision inherent in the logarithmic computation. It doesn't round or estimate, thus providing the full mathematical accuracy as it uses the logarithm in its exact form.

An approximation is an estimation of an exact solution, providing a numerical value rounded to a specific degree of precision. Approximations are useful for practical applications where exact values might be cumbersome or unnecessary.

With the equation \( e^{-0.08k} = 10 \), we calculate the approximation of \( k \) to four decimal places after finding the exact solution. This is done by computing \( \ln(10) \div -0.08 \).

The calculated value, which involves the natural logarithm, is

• \( k \approx -32.4278 \)

This approximation is useful in real-world scenarios and calculations where detailed precision is not crucial or when communicating the value in an easily interpretable form.

The step-by-step approach in solving equations breaks down a potentially complicated problem into simpler, more manageable tasks. This method ensures clarity and helps in understanding the underlying mechanics of solving equations.

For the given exponential equation \( e^{-0.08k} = 10 \), the step-by-step solution involves:

• Taking the natural logarithm of both sides to eliminate the exponent, \( \ln(e^{-0.08k}) = \ln(10) \)
• Utilizing the property of logarithms, \( (-0.08k)(1) = \ln(10) \)
• Isolating \( k \) by dividing each side by \(-0.08\), leading to the exact solution \( k = \frac{\ln(10)}{-0.08} \)
• Finally, approximating \( k \) to four decimal places to derive \( k \approx -32.4278 \)

This structured process not only leads to the correct answer but also builds a strong foundation in understanding exponential equations and their solutions.