What are the exact coordinates of the point of tangency?

Favorite Answer

Assume that the xy-coordinates of the point is (p,q) . Of course , q=2^p .
(d/dx)2^x = (ln2)2^x , and you said the tangent is the line y =mx , so
q = mp
2^p = ((ln2)2^p)p
1 = (ln2)p
p = 1 / ln2
q = 2^p = 2^(1 / ln2)
ln(q) = ln(2^(1 / ln2))
....... = (1 / ln2)ln2
....... = 1
So q = e ---> the xy-coordinates are (1 / ln2 , e)...Show more

y = mx has a slope of m

y = 2^x
y' = (2^x) * ln(2)

m = ln(2) * 2^(x)
m/ln(2) = 2^x
ln(m / ln(2)) = x * ln(2)
x = (1/ln(2)) * ln(m / ln(2))

y = mx
y = (m/ln(2)) * ln(m / ln(2))

((1/ln(2)) * ln(m / ln(2)) , (m / ln(2)) * ln(m / ln(2)))

There's your point....Show more