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Question

Question: How do you simplify \( \ln {e^4} \) ?...

How do you simplify lne4\ln {e^4} ?

Explanation

Solution

Hint : First we will convert this equation into the form logab{\log _a}b . Then we will evaluate all the required terms. Then we will apply the property. Here, we are using logab=bloga\log {a^b} = b\log a logarithmic property. The value of the logarithmic function lne\ln e is 11 .

Complete step-by-step answer :
We will first apply the logarithmic property to convert the equation to solvable form. Compare the given equation with formula and evaluate the values of the terms.
Here, the values are:
a=e b=4   a = e \\\ b = 4 \;
Hence, the equation will become:
lne4=4lne\ln {e^4} = 4\ln e
As, we know that lnee=1{\ln _e}e = 1 .
Therefore, the equation will become,
lne4=4lne lne4=4×1 lne4=4 {\ln {e^4}} = {4\ln e} \\\ {\ln {e^4}} = {4 \times 1} \\\ {\ln {e^4}} = 4
Hence, the simplified form of lne4\ln {e^4} will be 44 .
So, the correct answer is “4”.

Note : Remember the logarithmic property precisely which is logab=bloga\log {a^b} = b\log a .
While comparing the terms, be cautious. After the application of property when you get the final answer, tress back the problem and see if it returns the same values. Evaluate the base and the argument carefully. Also, remember that lnee=1{\ln _e}e = 1 .