Simplifying ln(aⁿ) using logarithmic rules | A comprehensive guide

ln(aⁿ)

To evaluate ln(aⁿ), where a is a positive number and n is any real number, we can apply the logarithm rules

To evaluate ln(aⁿ), where a is a positive number and n is any real number, we can apply the logarithm rules.

Firstly, we can rewrite aⁿ as (e^ln(a))ⁿ. This step allows us to convert the exponential expression to a logarithmic expression.

Using the logarithmic rule that ln(x^y) = y * ln(x), we can rewrite the expression as n * ln(e^ln(a)).

Now, we know that ln(e^x) = x, so we can simplify the expression further:

n * ln(e^ln(a)) = n * ln(a).

Hence, ln(aⁿ) simplifies to n * ln(a).

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