Finding all solutions to 3tan(ln x) = 2

[Nyasha]

Homework Statement

Find all solutions to the following equation: $$3tan(Inx)=2$$

The Attempt at a Solution

$$tan(Inx)=2/3$$

$$Inx=arctan(2/3)$$

x=e^(arctan(2/3)

 

[lanedance]

Hi Nyasha
i think you mean ln(x) = arctan(2/3)?

 

[lanedance]

note the potential for multiple solutions

 

[Nyasha]

Hi Nyasha
i think you mean ln(x) = arctan(2/3)?

Yes that's what l meant and my answer is x=e^(arctan(2/3). How do you get multiple solutions ?

 

[lanedance]

think about a graph of the function
y = tanx
it basically repeats along the axis every 2.pi

when you take the arctan, you can think of it as picking a y value, tracing it out to across to where it intersects the curve, and dropping down to the x value, giving you
x = arctan(y)

how do you choose a curve to use...? multiple solutions... how many are there?

 

[Nyasha]

think about a graph of the function
y = tanx
it basically repeats along the axis every 2.pi

when you take the arctan, you can think of it as picking a y value, tracing it out to across to where it intersects the curve, and dropping down to the x value, giving you
x = arctan(y)

how do you choose a curve to use...? multiple solutions... how many are there?

Are you trying to say it has multiple solutions because the domain of arctan is from -∞ to ∞

 

[lanedance]

no, to make it single valued, you must confine the range

think of your diagram of y = tanx
a vertical line will one graph
A horizontal will intersect many

try drawing y = arctanx
what does it look like? will essentially be your prvious graph rotated by 90degreees