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Chapter 12: Problem 13

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

Short Answer

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

Step by step solution

01

Express each side as a power of base 3

The equation is \(3^{1-x}=\frac{1}{27}\). We can write \(27\) as \(3^3\), so \(\frac{1}{3^3}\) can be written as \(3^{-3}\). Therefore, the equation now looks like this: \(3^{1-x}=3^{-3}\).
02

Equate the exponents

Since the bases on each side are equal (base 3), then the exponents must be equal too. This gives us the equation \(1-x = -3\).
03

Solve for x

Now we need to solve for \(x\). We can start by subtracting \(1\) from each side to isolate \(-x\), which gives \(-x = -4\). Multiplying each side by \(-1\) to solve for \(x\) gives us \(x = 4\).

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