Exponential Equations

An [b]exponential equation[/b] is one that has exponential expressions, in other words, powers that have in their exponent expressions with the unknown factor [i]x[/i]. For example, [br][br][img]https://www.matesfacil.com/ESO/exponenciales/expo8.png[/img][br][br]In this paper, we will resolve the exponential equations without using logarithms. This method of resolution consists in reaching an equality of the exponentials with the same base in order to equal the exponents.[br]

1. Remember...

Before we start, let's remember the properties of powers:[br][br][center][b]Product[br][br][img]https://www.matesfacil.com/ESO/potencias/potenciaDEF1.png[/img][br][br]Quocient [br][br][img]https://www.matesfacil.com/ESO/potencias/potenciaDEF3.png[/img][br][br]Inverse[br][br][img]https://www.matesfacil.com/ESO/potencias/potenciaDEF5.png[/img][br][br]Power[br][br][img]https://www.matesfacil.com/ESO/potencias/potenciaDEF2.png[/img][br][br]Negative exponent [br][br][img]https://www.matesfacil.com/ESO/potencias/potenciaDEF4.png[/img][br]Inverse of inverse [/b][br][br][img]https://www.matesfacil.com/ESO/potencias/potenciaDEF6.png[/img][/center]

2. Solved equations

Equation 1

[img]https://www.matesfacil.com/ESO/exponenciales/expo1.png[/img][br]Taking into account that [math]27=3^3[/math], we can rewrite the equation as[br][br][img]https://www.matesfacil.com/ESO/exponenciales/expo1-2.png[/img][br]Therefore, the solution is [math]x=3[/math].

Equation 2

[img]https://www.matesfacil.com/ESO/exponenciales/expo2.png[/img][br]Taking into account that [math]16=2^4[/math], we can rewrite the equation as[br][br][img]https://www.matesfacil.com/ESO/exponenciales/expo2-2.png[/img][br]Then we have the linear equation [math]x+2=4[/math]. Therefor, the solution is [math]x=2[/math].

Equation 3

[img]https://www.matesfacil.com/ESO/exponenciales/expo5.png[/img][br]Taking into account that[br][br][img]https://www.matesfacil.com/ESO/exponenciales/expo5-1.png[/img][br]We can rewrite the equation as[br][br][img]https://www.matesfacil.com/ESO/exponenciales/expo5-2.png[/img][br]We have the common base [math]3^x[/math], but because one of them is squared, we write[br][br][img]https://www.matesfacil.com/ESO/exponenciales/expo5-3.png[/img][br]Substituting, the equation finishes like[br][br][img]https://www.matesfacil.com/ESO/exponenciales/expo5-4.png[/img][br]In other words, a quadratic equation:[br][br][img]https://www.matesfacil.com/ESO/exponenciales/expo5-5.png[/img][br][br]We multiply the full equation by 9:[br][br][img]https://www.matesfacil.com/ESO/exponenciales/expo5-6.png[/img][br][br]We solve it:[br][br][img]https://www.matesfacil.com/ESO/exponenciales/expo5-7.png[/img][br]Therefore,[br][br][img]https://www.matesfacil.com/ESO/exponenciales/expo5-8.png[/img][br][br]So, we obtain[br][br][img]https://www.matesfacil.com/ESO/exponenciales/expo5-9.png[/img][br][br]The second option is not possible because it is negative. Therefore,[br][br][img]https://www.matesfacil.com/ESO/exponenciales/expo5-10.png[/img][br][br]From where we obtain the only one solution [math]x=2[/math].

More examples:

[list][br][*][url=https://www.matesfacil.com/english/high/solved-exercises-exponential-equations.html]Resolved exponential equations[/url][br][*][url=https://www.matesfacil.com/english/]Index of math content[/url][br][/list]

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