# Math Help - point of intersection

1. ## point of intersection

How the find the point of intersection of the two functions: y=300(1.05)^x and y=1000(0.92)^x? Help, Thanks

2. Originally Posted by Hellooo
How the find the point of intersection of the two functions: y=300(1.05)^x and y=1000(0.92)^x? Help, Thanks
I haven't looked to ensure that there exists a unique intersection point, but you can find any intersection points by solving

300(1.05)^x = 1000(0.92)^x

3. Hello,
you can let a = 1.05 and b = 0.92 to make it simpler, then you have to solve :

300 a^x = 1000 b^x

Taking the logarithms of base "a" yields :

log_a(300 a^x) = log_a(1000 b^x)

Simplifying using the log laws :

log_a(300) + log_a(a^x) = log_a(1000) + log_a(b^x)

Keep simplifying :

log_a(300) + x = log_a(1000) + x log_a(b)

Isolate the x's on one side :

log_a(300) - log_a(1000) = x log_a(b) - x

Simplify :

log_a(300 / 1000) = x(log_a(b) - 1)

Alright :

(log_a(0.3)) / (log_a(b) - 1) = x

Now replace a and b with the values defined at the start, do the calculation and you got x (provided it exists, you'll know it doesn't if you get a math error )

EDIT : sorry, used 100 instead of 1000, corrected and checked against Wolfram

4. $300(1.05)^x = 1000(0.92)^x$

$(1.05)^x / (0.92)^x = 10/3
$

$(1.05/0.92) ^x = 10/3$

$x log(1.05/0.92) = log(10/3)$

$x = log(10/3) / log(1.05/0.92)$

$= 9.109152249....$

5. Thank you very much!