Answer :
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]
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]
