Question

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

Answer 1

`(1)/(11)`

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^{(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^{(1)/(11) }`

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

`log_(e)(e^{(1)/(11) } ) = (1)/(11)`

Answer 2

1/11

Explanation:

We are asked to find the natural log of

`\sqrt[11]{e}`

Convert to fractional exponent

`ln(e {}^{ (1)/(11) } )`

Apply Log of Power rule

`(1)/(11) ln(e)`

Natural log of e is 1 so

`(1)/(11) * 1 = (1)/(11)`