how would you solve this? ps what is this type of equation called besides solving for x Thanks!

Question

9^2x=27^4 i wwant ro find a method to solve it besides guess and check (i know the answer is 3 but i used guess and check)

Answer 1

This is called a logarithmic equation because you are solving for the unknown exponential value.

 

To solve for x, we need to have a common base on both sides of the equation.  Notice that 3³  is 27.  3² is 9.

 

 

(3²)^2x = (3³)⁴

 

 

Using the rules of exponents,

 

3^4x = 3¹²

 

 

Now just equate the exponents.

 

4x = 12

 

 

Solve for x.  I leave this part to you.

 

Answer 2

The equation is an exponential equation.  The strategy to use is to take the logarithm of both sides or, if possible, rewrite each side as a power of the same base.  

 

9^2x = 27⁴

 

(3²)^2x = (3³)⁴

 

3^4x = 3¹²

 

So, 4x = 12

 

       x = 3