What function cancels out the natural logarithm (ln)?

What function cancels out the natural logarithm (ln)?

• A. Logarithmic functions
• B. Exponential function (e)
• C. Square root function
• D. Trigonometric functions

Answer: (B)

The exponential function \( e^x \) effectively cancels out the natural logarithm \( \ln(x) \) due to their inherent mathematical relationship. The natural logarithm is defined as the inverse function of the exponential function with base \( e \). This means that if you take the natural logarithm of \( e^x \), you will return to the value of \( x \): \[ \ln(e^x) = x \] Conversely, when you apply the exponential function to the natural logarithm, the result returns to the original argument of the logarithm: \[ e^{\ln(x)} = x \] This one-to-one relationship shows that the exponential function and the natural logarithm undo each other, allowing for the transformation of values between exponential and logarithmic forms. Regarding the other options, while logarithmic functions in general relate to various bases, they do not specifically cancel out the natural logarithm as directly as the exponential function does. The square root function serves a different purpose in mathematics and does not have a direct relationship with logarithmic functions. Trigonometric functions also do not interact with logarithms to cancel them out; instead, they are used primarily for circular functions and angles. These distinctions

The exponential function ( e^x ) effectively cancels out the natural logarithm ( \ln(x) ) due to their inherent mathematical relationship. The natural logarithm is defined as the inverse function of the exponential function with base ( e ). This means that if you take the natural logarithm of ( e^x ), you will return to the value of ( x ):

[

\ln(e^x) = x

]

Conversely, when you apply the exponential function to the natural logarithm, the result returns to the original argument of the logarithm:

[

e^{\ln(x)} = x

]

This one-to-one relationship shows that the exponential function and the natural logarithm undo each other, allowing for the transformation of values between exponential and logarithmic forms.

Regarding the other options, while logarithmic functions in general relate to various bases, they do not specifically cancel out the natural logarithm as directly as the exponential function does. The square root function serves a different purpose in mathematics and does not have a direct relationship with logarithmic functions. Trigonometric functions also do not interact with logarithms to cancel them out; instead, they are used primarily for circular functions and angles. These distinctions