SOLUTION: Determine the point(s) of intersection of {{{y=2(3^x)}}} and {{{y=6(2^x)}}} algebraically. Round your answer to one decimal place.

Question 1205560: Determine the point(s) of intersection of and algebraically. Round your answer to one decimal place.
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
you have two equations that you want to solve simultaneously (the same value of x applies to both equations).

the equations are:
y = 2*3^x and y = 6*2^x.

set them equal to each other to get:
2*3^x = 6*2^x

divide both sides of the equation by 2 to get:
3^x = 3*2^x

divide both sides of the equation by 2^X to get:
3^x / 2^x = 3

take the log of both dies of the equqtion to get:

log(3^x / 2^x) = log(3)

since log(a/b) = log(a) - log(b), you get:

log(3^x) - log(2^x) = log(3)

since log(a^b) = b*log(a), you get:

x * log(3) - x * log(2) = log(3)

factor out the x to get:

x * (log(3) - log(2) = log(3)

solve for x to get:

x = log(3) / (log(3) - log(2)) = 2.709511291.

the graph of both equations will intersect when the value of x is 2.709511291.

the value of y at that point will be 39.24600209.

here's the graph.

here's a reference on the properties of logs (also known as log rules).

https://www.chilimath.com/lessons/advanced-algebra/logarithm-rules/
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