Question

How do you solve `10^ { x } \cdot e ^ { x } = 3`?

Answer 1

`x=(ln|3|)/(ln|10|+1)~~0.33265`

Explanation:

Take the natural log of both sides

`ln|10^xe^x|=ln|3|`

Using the following property of logarithms
we can rewrite the left hand side

`ln|xy|=ln|x|+ln|y|`

`ln|10^x|+ln|e^x|=ln|3|`

Using another property of logarithms we
can rewrite the left hand side again

`ln|x^y|=yln|x|`

`xln|10|+xln|e|=ln|3|`

Recall that `ln|e|=1`

`xln|10|+x=ln|3|`

Factor out an `x` on the left

`x(ln|10|+1)=ln|3|`

Solve for `x`

`x=(ln|3|)/(ln|10|+1)~~0.33265`