Question 1205560
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.


<img src = "http://theo.x10hosting.com/2024/010801.jpg">


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


<a href = "https://www.chilimath.com/lessons/advanced-algebra/logarithm-rules/" target = "_blank">https://www.chilimath.com/lessons/advanced-algebra/logarithm-rules/</a>