# Graphing utility question with 2 plots - one line and one exponential - is there a trick to this?

## In trying to do a recent question (find solutions to ${5}^{3} - 2 x = {5}^{- x}$), I tried putting both the left and right onto a single graph: (y-125+2x)(y-5^(-x))=0 and the resulting graph was... odd. Is there some trick to using the graphing utility when working with exponentials?

Dec 30, 2017

The reason is kind of funny.

#### Explanation:

The number 125 rises the line really high up, beyond deault range of graph.

graph{(y-125+2x)(y-5^-x)=0 [-10, 10, -5, 5]} graph{(y-125+2x)(y-5^-x)=0 [-10, 10, -5, 5]} To see it we must zoom out to include points of intersection with axes (0, 125)$\mathmr{and}$(62.5, 0). Keep the ratio if you need to.

graph{(y-125+2x)(y-5^-x)=0 [-200, 200, -20, 180]} graph{(y-125+2x)(y-5^-x)=0 [-200, 200, -20, 180]} Now we see the exponent and the line. Also something interesting happens at x~~-20. Line starts to oscillate rapidly. Let's zoom in:

graph{(y-125+2x)(y-5^-x)=0 [-35, -15, 165, 175]} graph{(y-125+2x)(y-5^-x)=0 [-35, -15, 165, 175]} Yep. Looks like floating point error. At x=-22 the value of exponent becomes to big for graphing utility and it affects the line too.

You can

• cut the domain by graphing (y-125+2x)(y-5^-x)sqrt(x+20)=0 More about cutting domains
• change range to hide it.
• logarithm both sides to avoid extremely high values (probably the best solution, but cuts off negative $y$)

graph{(y-125+2x)(lny+xln5)=0 [-200, 200, -20, 180]}
graph{(y-125+2x)(lny+xln5)=0 [-200, 200, -20, 180]}