Verified Answer :

  • Identify the logarithm power property: \(\log_b(m^n) = n \cdot \log_b(m)\).
  • Recognize the components in \(\log_6 e^f\): base \(b = 6\), argument \(m = e\), and exponent \(n = f\).
  • Substitute these into the property to obtain \(f \cdot \log_6 e\).
  • The correct equivalent expression is \(\boxed{f \log_6 e}\).

Explanation

  1. Analyzing the Logarithm Power Property

The problem asks us to rewrite the logarithmic expression \(\log_6 e^f\) using the logarithm power property. The logarithm power property states that for any base \(b > 0\) (with \(b eq 1\)) and any positive real number \(m\), the expression \(\log_b(m^n)\) can be rewritten as \(n \cdot \log_b(m)\).

  1. Applying the Property to the Expression

In our given expression \(\log_6 e^f\), we identify the components:

  • The base \(b\) is \(6\).
  • The argument \(m\) is \(e\).
  • The exponent \(n\) is \(f\).

Applying the property \(\log_b(m^n) = n \cdot \log_b(m)\), we substitute these values into the formula to get \(f \cdot \log_6 e\).

  1. Conclusion and Selection

Comparing our result \(f \cdot \log_6 e\) with the provided options:

  • \(e \log_6 f\)
  • \(f \log_6 e\)
  • \(\log_6(e \cdot f)\)
  • \(\log_6(e+f)\)

We can clearly see that our calculated expression \(f \log_6 e\) matches the second option. Thus, the expression \(\log_6 e^f\) is equivalent to \(f \log_6 e\), and the final answer is \(\boxed{f \log_6 e}\).

Examples

The logarithm power property is essential for simplifying complex equations in scientific fields. For example, in chemistry, the pH of a solution is calculated using the negative logarithm of the hydrogen ion concentration, \(\text{pH} = -\log[H^+]\). When dealing with exponential growth models, such as compound interest or population dynamics defined by \(A = P(1+r)^t\), taking the logarithm of both sides allows us to bring the time variable \(t\) out of the exponent: \(\log(A) = \log(P) + t \log(1+r)\). This makes the equation much easier to solve algebraically.

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