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How to Put Ln Tan 2 X in Graphing Calculator

Reviewed by Calculator Editorial Team

Graphing the natural logarithm of tangent of 2x (ln(tan(2x))) requires careful consideration of the function's domain and periodicity. This guide provides step-by-step instructions for entering and graphing this function in a graphing calculator.

How to Graph ln(tan(2x))

The function ln(tan(2x)) combines the logarithmic and trigonometric functions. To graph it properly, you need to understand its behavior and how to enter it in your calculator.

Function: y = ln(tan(2x))

Domain: x ≠ (2n + 1)π/4 for any integer n, where tan(2x) ≠ 0

Period: π/2 (the function repeats every π/2 units)

The function is undefined where tan(2x) is zero or negative because the natural logarithm is only defined for positive real numbers. The vertical asymptotes occur where tan(2x) approaches infinity or zero.

Step-by-Step Instructions

  1. Enter the Function

    On your graphing calculator, enter the function as Y1 = ln(tan(2x)). The exact syntax may vary slightly depending on your calculator model.

  2. Set the Window

    Adjust the window settings to view the function clearly. A good starting point is:

    • Xmin: -π
    • Xmax: π
    • Xscl: π/4
    • Ymin: -2
    • Ymax: 2
    • Yscl: 0.5

  3. Graph the Function

    Press the graph button to display the function. You should see a series of curves with vertical asymptotes where the function is undefined.

  4. Adjust for Clarity

    If the graph is too crowded, adjust the window settings to focus on specific intervals where the function is defined and continuous.

Common Issues and Solutions

Issue: Calculator shows "Error" or "Undefined"

This typically occurs when you try to evaluate the function at points where tan(2x) is zero or negative. Adjust your window settings to avoid these points.

Issue: Graph looks distorted or incomplete

Check your window settings and ensure you're not zoomed in too far. The function has a period of π/2, so you may need to adjust the Xmin and Xmax values to see a complete cycle.

Example Calculation

Let's evaluate ln(tan(2x)) at x = π/8:

  1. Calculate tan(2 * π/8) = tan(π/4) = 1
  2. Take the natural logarithm: ln(1) = 0

So, ln(tan(2 * π/8)) = 0. This point will appear on the graph as a smooth transition between the positive and negative parts of the function.

FAQ

Why does the graph have vertical asymptotes?
Vertical asymptotes occur where tan(2x) approaches infinity or zero. The natural logarithm is undefined for non-positive numbers, so these points create vertical asymptotes in the graph.
How do I find the period of ln(tan(2x))?
The period of tan(2x) is π/2, and the natural logarithm doesn't affect the periodicity. Therefore, ln(tan(2x)) has the same period of π/2.
Can I graph ln(tan(2x)) on my smartphone?
Yes, many graphing apps for smartphones support logarithmic and trigonometric functions. Follow similar steps as with a physical calculator.