Problem 93 - Algebra and Trigonometry

Question

Solve each equation. $$ 5^{2 x} \cdot 5^{4 x}=125 $$

Step-by-Step Solution

Answer

The solution to the equation is \(x = 0.5\).

Step 1: Combine the powers with the same base

Since the bases are the same (both are 5), the rule of multiplication of powers with the same base can be used. The rule states that \(a^{m} * a^{n} = a^{m+n}\). Here our base is 5, therefore we have \(5^{2x} * 5^{4x} = 5^{2x + 4x} = 5^{6x}\).

Step 2: Swap the left and right side of the equation

The equation now is \(5^{6x} = 125\). Swap the two sides for convenience. This doesn't affect the solution: \(125 = 5^{6x}\).

Step 3: Transfer 125 to base 5

To apply the principles of logarithms later, it is helpful to have the number 125 in the base of 5. The number 125 is actually \(5^3\), therefore, the equation now is \(5^3 = 5^{6x}\).

Step 4: Equate the powers of the equation

Now the bases are the same (i.e., base 5), we can equate the powers. This gives us the equation \(3 = 6x\).

Step 5: Solve for x

Finally, we can solve for \(x\) by dividing both sides by 6. This gives us \(x = 3/6 = 0.5\).