Solving exponential equations with x as the exponent

[Tyrion101]

My confusion comes from basic exponent rules and whether or not both sides of an equation have to have the same level of exponent, when you reduce the base for solving. If one side can have an exponent of 3, does the other side also have to be reduced to something that would also have an exponent of 3? 9^2 does equal 81, but is that wrong if the other side can be reduced to a 3rd power, and can't be for a 2nd? I hope I'm making sense.

 

[Mark44]

My confusion comes from basic exponent rules and whether or not both sides of an equation have to have the same level of exponent, when you reduce the base for solving. If one side can have an exponent of 3, does the other side also have to be reduced to something that would also have an exponent of 3? 9^2 does equal 81, but is that wrong if the other side can be reduced to a 3rd power, and can't be for a 2nd? I hope I'm making sense.

The exponents don't have to be the same, but the bases have to be the same if you're using this idea: a^m = aⁿ $\Rightarrow$ m = n.

 

[Tyrion101]

Ok I think I understand

 

[1MileCrash]

Just to nit pick, that is true when a =\= 1 and a =\= 0.

 

[CellCoree]

I can understand your confusion with solving exponential equations. Exponents can be tricky, but there are certain rules that can help us solve them correctly.

Firstly, it is important to remember that when solving an exponential equation, the goal is to isolate the variable (in this case, x) on one side of the equation. In order to do this, we need to use the properties of exponents to manipulate the equation.

One of the properties of exponents is the power rule, which states that when you have an exponent raised to another exponent, you can multiply the exponents together. For example, (x^2)^3 can be simplified to x^6.

In your example, 9^2 does equal 81, but this does not mean that the other side of the equation also has to have an exponent of 2. The key is to make sure that the exponents on both sides of the equation are equal after simplifying.

For instance, if one side of the equation has an exponent of 3, you can reduce the other side to a 3rd power as well. This could involve using the power rule or other exponent properties, such as the product rule or quotient rule.

In summary, the important thing to remember is that both sides of the equation do not have to have the same level of exponent, but they should have equal exponents after simplifying. I hope this helps to clarify your confusion.