Question

Solve for \(x\). $$ 3^{x}=81 $$

Short answer

The value of \(x\) that solves the equation \(3^x = 81\) is \(x = 4\).

Step-by-step solution

Express 81 as a power of 3

As stated, the key to solving an exponential equation is to have the base the same on both sides. In this case, the base on the left side of the equation is 3; so if we can express 81 as a power of 3 on the right, then we can equate and solve for \(x\). Now, \(3^4\) is 81, so we replace 81 with \(3^4\) in equation.

Equate the powers

Now the equation becomes \(3^x = 3^4\). Since the bases on both sides are the same (both are 3), we can then equate the powers and solve for \(x\). So, \(x\)=4.

Check solution

Don't forget to check if the obtained value of \(x\) is correct. Substitute \(x\) with 4 in the original equation: \(3^{x}=81\) becomes, \(3^4 = 81\), it returns true. Therefore, \(x\) = 4 is the solution to the given equation.