How do you find the equation of the tangent line to the curve y=ln(3x-5) at the point where x=3?

Feb 2, 2016

$y = \frac{3}{4} x - .864$

Explanation:

The equation of the tangent line is composed of two parts: the derivative (slope), and the $y$-intercept. The problem contains all the information necessary to solve it; we just have to be clever and use the right techniques.

First we find the derivative of our function at $x = 3$. This will be the slope of the tangent line. (The tangent line is straight, so the slope is the same at all points - whether it's $x = 3 , 4 , 5 , 6 , 7 ,$etc). The derivative of the natural log function is $\frac{1}{x}$, so we know the derivative of $\ln \left(3 x - 5\right)$ will be $\frac{1}{3 x - 5}$.

However, the chain rule tells us that we need to multiply this by the derivative of the "inside" function (in our case, $3 x - 5$) - which is $3$. Therefore, the complete derivative is
$y ' \left(x\right) = \frac{3}{3 x - 5}$.

To find the derivative at $x = 3$, we simply plug in $3$ for $x$:
$y ' \left(3\right) = \frac{3}{3 \left(3\right) - 5}$

$y ' \left(3\right) = \frac{3}{9 - 5}$

$y ' \left(3\right) = \frac{3}{4}$

Thus the slope of our tangent line is $\frac{3}{4}$. Now, finding the $y$-intercept requires the slope - which we have - and a point from the function $y = \ln \left(3 x - 5\right)$. Because we're evaluating the tangent line at $x = 3$, we will use $3$ to find our point:
$y \left(3\right) = \ln \left(3 \left(3\right) - 5\right)$
$y \left(3\right) = \ln \left(9 - 5\right)$
$y \left(3\right) = \ln \left(4\right)$

We can finally move on to the last step - finding the actual equation. Because we know the tangent line is straight, it will be of the form $y = m x + b$, where $m$ is the slope and $b$ is the $y$-intercept. We have everything we need:
$y = \ln \left(4\right) \to$ ($y$ part of the point)
$x = 3 \to$ ($x$ part of the point)
$m = \frac{3}{4} \to$ (slope)

Let's plug these numbers into our equation:
$\ln \left(4\right) = \left(\frac{3}{4}\right) \left(3\right) + b$
$\ln \left(4\right) = \frac{9}{4} + b$
$\ln \left(4\right) - \frac{9}{4} = b$

That's our $y$-intercept, making the equation of the tangent line $y = \frac{3}{4} x + \ln \left(4\right) - \frac{9}{4}$. If you prefer hard numbers instead of logs and fractions, this can be approximated to
$y = \frac{3}{4} x - .864$

See the graph below for a visual description of the solution.