Problem 20 - College Algebra Essentials

Question

Solve each exponential equation by expressing each side as a power of the same base and then equating exponents. $$8^{1-x}=4^{x+2}$$

Step-by-Step Solution

Answer

The solution to the equation is \(x = -1\).

Step 1: Express 8 and 4 as Powers of 2

The first step involves changing the bases to a common base. Both 8 and 4 can be written as powers of 2 such that \(8^{1-x}\) is \(2^{3*(1-x)}\) and \(4^{x+2}\) is \(2^{2*(x+2)}\). This changes the original equation to: \[2^{3*(1-x)}=2^{2*(x+2)}\]

Step 2: Equate the Exponents

The next step is to leverage the property of equality of bases implying equality of exponents, equating the exponents from the rewritten equation. This gives a new equations as: \[3*(1-x)=2*(x+2)\]

Step 3: Solve for x

Finally, the last step is to solve the equation for x. Start by expanding the right-hand side term: \[3 - 3x = 2x + 4\]. Then, you would isolate x by adding 3x to both sides and subtracting 4 from both sides. Solving that equation will give x = -1.