Question

Finding an Equation of a Tangent Line In Exercise, find an equation of the tangent line to the graph of the function at the given point. See Example 5.

y = log3 x; (27, 3)

Answer 1

Answer: y - 3 = 0.041(x - 27)

Explanation:

y = log3 x; (27, 3)

Step 1

Find the point of tangency.

It's given as (27,3)

Step 2

Find the first derivative, and evaluate it at x=27

Differentiating log3 x.

Note we can't directly differentiate log3 x, we need to convert it to natural log = ln.

Using change of base.

Log3 x = lnx/ln3

So,

d(lnx/ln3)/dx = 1/ln3 × 1/x

f'(x) = 1/xln3

f'(27) = 1/27ln3 = 0.041

The slope of the tangent line at this point is m= 0.041

Step 3

Find the equation of the tangent line at (27,3) with a slope of m=0.041

y−y1 = m(x - x1)

y - 3 = 0.041(x - 27)