Question

How do you write \[{{e}^{3}}=20.0855\] in logarithmic form?

Answer 1

Hint: In this problem, we have to convert the given exponential form into its logarithmic form. We can first take log on both the left-hand side and the right-hand side to remove the exponent form and to write in logarithmic form. We can then use the logarithmic formula or identity \[\ln e=1\] on the left-hand side, we will get the logarithmic form.

Complete step-by-step solution:
We know that the given exponential form is,
\[{{e}^{3}}=20.0855\]
We know that a natural exponential equation in logarithmic form, is the result of the exponential equation which becomes the argument of the natural logarithm and the exponent in the exponential expression becomes the result of the natural logarithm equation, we can say that \[{{e}^{A}}=B\Rightarrow \ln B=A\].
We can now take log on both the left-hand side and the right-hand side to remove the exponent form and to write in logarithmic form, we get
\[\Rightarrow \log {{e}^{3}}=\log 20.0855\]
We can now use the logarithmic identity, \[\ln e=1\] in the above step, we get
\[\Rightarrow 3=\log 20.0855\]
Therefore, the logarithmic form of the given exponential form \[{{e}^{3}}=20.0855\] is \[3=\log 20.0855\].

Note: We should always remember that a natural exponential equation in logarithmic form, is the result of the exponential equation which becomes the argument of the natural logarithm and the exponent in the exponential expression becomes the result of the natural logarithm equation, we can say that \[{{e}^{A}}=B\Rightarrow \ln B=A\]. We should also know some logarithmic formula or identity such as\[\ln e=1\].