Question

How do you write \[\ln 4=1.386\].. in exponential form?

Answer 1

Hint: The logarithm base b of a number n is the number x that when b is raised to \[{{x}^{th}}\] power, the resulting value is n. That is, this can be written as \[{{\log }_{b}}n=x\Leftrightarrow {{b}^{x}}=n\]. The logarithm with base \[e\] is called the natural logarithm representing the \[\ln \] function. So, we can write the logarithm base \[e\] of a number n is the number x which can be expressed in exponents as \[\ln n={{\log }_{e}}n=x\Leftrightarrow {{e}^{x}}=n\].

Complete step by step answer:
As per the given question, we are provided with a logarithmic expression. We need to express this logarithmic expression in exponential form. And the given logarithmic expression is \[\ln 4=1.386\]..
If we observe the given expression, we get to know that the base of this logarithmic function is \[e\] which is a natural logarithm.
From the definition of logarithm, we can say that the logarithm base \[e\] of a number 4 is equal to 1.386. That is, when \[e\] is raised to the power of 1.386, we get back the same number 4.
Thus, we can write the given logarithm expression \[\ln 4=1.386\].. in the exponential form as \[{{e}^{1.386}}=4\]. That is, we can write it as
\[\Rightarrow \ln 4=1.386\Leftrightarrow {{e}^{1.386}}=4\]

\[\therefore {{e}^{1.386}}=4\] is the required exponential form of \[\ln 4=1.386\].

Note: We can directly find the exponential form of a logarithmic expression by taking the image of the graph of the logarithmic function with respect to any of the lines \[y=\pm x\] based on the existence of the graph. We make mistakes with logarithms because we are working with the exponents in reverse. The mistakes go with the wrong implementation of the exponent and logarithm properties. We should be thorough with the logarithm and exponent concept to avoid mistakes like taking base 10 for \[\ln \] function instead of \[e\].