Find the exact value of the logarithm without using a calculator.
In
(Type an integer or a simplified fraction.)

Ask by Washington Chandler. in the United States

Mar 26,2025

Question

Find the exact value of the logarithm without using a calculator. In $$ \sqrt[8]{e} $$ $$ \ln \sqrt[8]{e}=\square $$ (Type an integer or a simplified fraction.)

UpStudy AI · Accepted Answer

1. Recognize that the eighth root of $$ e $$ can be written as an exponent:
   $$
   \sqrt[8]{e} = e^{\frac{1}{8}}.
   $$

2. Apply the natural logarithm:
   $$
   \ln \sqrt[8]{e} = \ln \left(e^{\frac{1}{8}}\right).
   $$

3. Use the logarithm property $$\ln (a^b) = b \ln (a)$$ with $$a = e$$ and $$b = \frac{1}{8}$$:
   $$
   \ln \left(e^{\frac{1}{8}}\right) = \frac{1}{8} \ln (e).
   $$

4. Since $$\ln (e) = 1$$, this simplifies to:
   $$
   \ln \sqrt[8]{e} = \frac{1}{8}.
   $$

Thus, the exact value is $$\frac{1}{8}$$.
$$\ln \sqrt[8]{e} = \frac{1}{8}$$

Find the exact value of the logarithm without using a calculator.
In
(Type an integer or a simplified fraction.)

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Find the exact value of the logarithm without using a calculator. In (Type an integer or a simplified fraction.)