Problem 7 - College Algebra Essentials

Question

Solve each exponential equation by expressing each side as a power of the same base and then equating exponents. $$4^{2 x-1}=64$$

Step-by-Step Solution

Answer

The solution to the equation \(4^{2 x-1}=64\) is \(x = 2\).

Step 1: Expressing both sides in the same base

First, we need to express 4 and 64 as powers of 2. The number 4 is equal to \(2^2\) and 64 is equal to \(2^6\). Now, substitute \(2^2\) for 4, and \(2^6\) for 64 in the given equation.\n Hence, the equation becomes: \n \((2^2)^{2 x-1} =2^6\)

Step 2: Simplifying the LHS using the power of a power rule

The power of a power rule states that \((a^m)^n = a^{m*n}\). Applying this rule to the left hand side (LHS) of the equation, we get: \n \( => (2^{2*(2x-1)}) = 2^6 \) or \(2^{4x-2}=2^6\).

Step 3: Equating exponents

Since the bases are the same, the exponents must be equal. So, set 4x - 2 equal to 6. \n==> 4x - 2 = 6

Step 4: Solve for \(x\)

Solving the equation for \(x\), we get: \n 4x = 6 + 2\n 4x = 8 \n Thus, dividing both sides by 4, we get \(x = 8/4 = 2\).