# How to solve ln (1+x)= 1+ ln x ?

May 23, 2018

$x = \frac{1}{e - 1} \approx 0.582$

#### Explanation:

Step 1
First, we must move all terms to one side.
$\ln \left(1 + x\right) - 1 - \ln x = 0$

Step 2
We can now further simplify using the quotient rule.
$\ln \left(\frac{1 + x}{x}\right) - 1 = 0$

Step 3
We can now combine like terms to reduce the equation.
$\ln \left(\frac{1}{x} + 1\right) - 1 = 0$

Step 4
Next, we begin to isolate the variable, $x$, by moving everything else to the other side.
$\ln \left(\frac{1}{x} + 1\right) = 1$

Step 5
We then use the natural logarithm.
$\frac{1}{x} + 1 = e$

Step 6
We continue to isolate $x$.
$\frac{1}{x} = e - 1$

Step 7
Finally, we simplify and determine our final answer.
$x = \frac{1}{e - 1} \approx 0.582$