Problem 13 - College Algebra Essentials

Question

Solve each exponential equation by expressing each side as a power of the same base and then equating exponents. $$3^{1-x}=\frac{1}{27}$$

Step-by-Step Solution

Answer

Therefore, the solution to the equation \(3^{1-x}=\frac{1}{27}\) is \(x=4\).

Step 1: Express both sides using the same base

In order to solve an exponential equation, one of the methods employed is to express both sides of the equation with the same base. In this equation, notice that 27 can be expressed as \(3^3\), and 1 divided by any number can be represented as that number to the power of -1. Therefore, the equation \(3^{1-x}=\frac{1}{27}\) can also be written as \(3^{1-x}=3^{-3}\).

Step 2: Equate the exponents

The next step is to set the exponents equal to each other as both sides of the equation have the same base: \(1-x=-3\).

Step 3: Solve the resulting equation

Once the equation \(1-x=-3\) is obtained, it can be solved for x by first subtracting 1 from both sides which yields \(-x=-4\), and then by multiplying both sides by -1 which yields \(x=4\).