Question

Solve for x in the exponential equation below and check your solution.27x-2 = 81*-1

Answer 1

To be able to solve the given exponential equation, let's make sure first that they will have the same base.

As of now, their bases are 27 and 81. Let's rewrite these two numbers in such a way that they will have the same base.

`\begin{gathered} 27=3^3 \\ 81=3^4 \end{gathered}`

From this, we can rewrite the exponential equation into:

`3^(3(x-2))=3^(4(x-1))`

Since both sides now have the same base, we can now equate the exponents of each base.

`3(x-2)=4(x-1)`

Let's simplify the equations on both sides by distributing 3 and 4 to the terms inside each respective parenthesis.

`3x-6=4x-4`

Let's subtract 3x on both sides of the equation.

`\begin{gathered} 3x-6-3x=4x-4-3x \\ -6=x-4 \end{gathered}`

Let's add 4 on both sides of the equation.

`\begin{gathered} -6+4=x-4+4 \\ -2=x \end{gathered}`

Therefore, the value of x is -2.

`\begin{gathered} 27^(-2-2)=81^(-2-1) \\ 27^(-4)=81^(-3) \\ (1)/(27^4)=(1)/(81^3) \\ (1)/(531,441)=(1)/(531,441) \end{gathered}`