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Question 1 of 25

Which logarithmic equation is equivalent to the exponential equation below?

[tex]\[ e^{4x} = 5 \][/tex]

A. \(\log 4x = 5\)
B. \(\ln 4x = 5\)
C. \(\ln 5 = 4x\)
D. [tex]\(\log 5 = 4x\)[/tex]


Sagot :

To solve the equation \( e^{4x} = 5 \) for \( x \), we can use logarithms to simplify the exponential form. Here is the step-by-step process:

1. Identify the exponential equation:
[tex]\[ e^{4x} = 5 \][/tex]

2. Take the natural logarithm of both sides:
[tex]\[ \ln(e^{4x}) = \ln(5) \][/tex]

3. Apply the property of logarithms: The natural logarithm of an exponential function \( \ln(e^y) \) simplifies to \( y \). In other words, \( \ln(e^y) = y \). Thus,
[tex]\[ 4x \cdot \ln(e) = \ln(5) \][/tex]

4. Since \(\ln(e) = 1\) (because the natural logarithm of \(e\) is 1 by definition), we simplify the equation to:
[tex]\[ 4x \cdot 1 = \ln(5) \][/tex]
[tex]\[ 4x = \ln(5) \][/tex]

5. Therefore, the logarithmic equation equivalent to the exponential equation \( e^{4x} = 5 \) is:
[tex]\[ \ln(5) = 4x \][/tex]

Comparing this with the given options, we see that option C corresponds to this form:
[tex]\[ \ln 5 = 4x \][/tex]

Hence, the correct answer is:
[tex]\[ \boxed{\text{C. } \ln 5 = 4x} \][/tex]