How can I solve the equation 6^x = 1296 by expressing both sides as a power of the same base?

To solve the equation 6^x = 1296, we first need to express both sides as powers of the same base.

Let’s start with 1296. We can find the prime factorization of 1296. By dividing 1296 by 6:

• 1296 ÷ 6 = 216
• 216 ÷ 6 = 36
• 36 ÷ 6 = 6
• 6 ÷ 6 = 1

From the above divisions, we see that 1296 can be expressed as:

6 × 6 × 6 × 6 = 6⁴

Hence, we can rewrite the equation as:

6^x = 6⁴

Now that both sides of the equation have the same base (which is 6), we can equate the exponents:

x = 4

Thus, the solution to the equation 6^x = 1296 is:

x = 4