Question

How do you solve `lnx=5-ln2x`?

Answer 1

`x=8.614`, rounded to three decimal places.

Explanation:

To solve `lnx=5-ln2x`, for `x`

Rearranging to bring all terms containing `x` to left hand side of the equation we obtain
`lnx+ln2x=5`
Using the property of logarithms that `log (axxb)=loga+log b` we obtain
`ln(x xx 2x)=5`
or `ln2 x^2=5`

Using the definition of log functions we obtain

`e^5=2x^2`, solving for `x`
`x=+-sqrt(e^5/2)`, ignoring the `-ve` sign as logarithms of a negative number is not defined.

`x=sqrt(e^5/2)`, with a calculator

or `x=8.614`, rounded to three decimal places.