Transcendent equations

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Transcendent equations are a type of equations. In this case, they are those that cannot be reduced to an equation, of the form f (x) = 0, to solve through algebraic operations.

That is, transcendent equations cannot be easily solved with addition, subtraction, multiplication, or division. However, the value of the unknown can sometimes be found using analogies and logic (we'll see with examples later).

A common feature of transcendent equations is that they often have bases and exponents on both sides of the equation. Thus, to find the value of the unknown, the equation can be transformed, looking for the bases to be equal, and, in this way, the exponents can also be equal.

Another way to solve transcendent equations, if the exponents of both sides are similar, is by equating the bases. Otherwise, you can look for other similarities (this will become clearer with an example that we will show later).

Difference between transcendent equations and algebraic equations

Transcendental equations differ from algebraic equations in that the latter can be reduced to a polynomial equal to zero, of which, later, their roots or solutions can be found.

However, transcendent equations, as mentioned above, cannot be reduced to the form f (x) to be solved.

Examples of transcendent equations

Let's see some examples of transcendent equations and their solution:

Example 1

  • 223 + 8x = 42-6x

In this case, we transform the right side of the equation to have equal bases:

223 + 8x = 22 (2-6x)

223 + 8x = 24-12x

Since the bases are equal, we can now equal the exponents:

23 + 8x = 4-12x

20x = -19

x = -0.95

Example 2

  • (x + 35) a = (4x-16) 2a

In this example, it is possible to equalize the bases and solve for the unknown x.

(x + 35) a = ((4x-16) 2) a

x + 35 = (4x-16) 2

x + 35 = 16x2-128x + 256

16x2-129x-221 = 0

This quadratic equation has two solutions following the following formulas, where a = 16, b = -129 and c = -221:

Then,

Example 3

  • 4096 = (x + 2) x + 4

We can transform the left side of the equation:

46 = (x + 2) x + 4

Therefore, x is equal to 2, and it is true that the base is x + 2, that is, 4, while the exponent is x + 4, that is, 6.

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