The equation of a line perpendicular to the curve $y = \log_e (2x − 1)$ has the equation $y = −2x + k$, where $k$ is a constant. Find the value of $k$, correct to $1$ decimal place.
I know that a perpendicular line has a gradient of $-1/m$ to the line that it is perpendicular with, but other than that I'm not even sure where to start with this question. Any suggestions would be appreciated. Thanks
We are asked to find the point on the curve, where slope of normal is $-2$. We know that slope of normal is $-2$ if and only if slope of the tangent at that point is $\frac{1}{2}$. Now, $$\frac{dy}{dx}=\frac{2}{2x-1}$$ Hence, at the solution point, $x=\frac{5}{2}$. Hence, solving for $y$ coordinate, $$k-5=\ln{4}$$ Now, find the value of $k$ in decimals.
Hope it helps:)