Question

Find the exact value of the logarithm without using a calculator.

Answer 1

Answer:

$\frac{1}{11}$

Step-by-step explanation:

First, remember that the ln function is just a log function with a base of e. Here's how it looks

$ln(x) =log_{e}(x)$

$ln(\sqrt[11]{e} ) = log_{e}(\sqrt[11]{e} )$

We can take this one step further if we realize that we can rewrite the square root as a simple power to a fraction!

$log_{e}(e^{\frac{1}{11} } )$

Solving the equation above is really simple. All that function is really saying is can we raise e to some number, where the result would be e^(1/11)? In other words find x.

$e^{x} = e^{\frac{1}{11} }$

Well x has to be 1/11 in that case. And that ends up being our final answer.

$log_{e}(e^{\frac{1}{11} } ) = \frac{1}{11}$

Answer 2

Answer:

1/11

Step-by-step explanation:

We are asked to find the natural log of

$\sqrt[11]{e}$

Convert to fractional exponent

$ln(e {}^{ \frac{1}{11} } )$

Apply Log of Power rule

$\frac{1}{11} ln(e)$

Natural log of e is 1 so

$\frac{1}{11} \times 1 = \frac{1}{11}$