Absolutely! Let's solve the equation [tex]\(7^{2x+3} = 343^x\)[/tex] step by step.
First, recognize that [tex]\(343\)[/tex] can be expressed as a power of [tex]\(7\)[/tex]:
1. Notice that [tex]\(343 = 7^3\)[/tex].
So, we can rewrite the equation [tex]\(7^{2x+3} = 343^x\)[/tex] using [tex]\(343 = 7^3\)[/tex]:
2. Substitute [tex]\(343\)[/tex] with [tex]\(7^3\)[/tex]:
[tex]\[7^{2x+3} = (7^3)^x\][/tex]
3. Simplify the right-hand side using the laws of exponents:
[tex]\[(7^3)^x = 7^{3x}\][/tex]
4. Now, our equation looks like this:
[tex]\[7^{2x+3} = 7^{3x}\][/tex]
5. Since the bases are the same (both are base 7), we can set the exponents equal to each other:
[tex]\[2x + 3 = 3x\][/tex]
6. Now, solve for [tex]\(x\)[/tex]:
[tex]\[2x + 3 = 3x\][/tex]
Subtract [tex]\(2x\)[/tex] from both sides:
[tex]\[3 = 3x - 2x\][/tex]
[tex]\[3 = x\][/tex]
So, the solution to the equation [tex]\(7^{2x+3} = 343^x\)[/tex] is:
[tex]\[x = 3\][/tex]
Therefore, [tex]\(x = 3\)[/tex] is our answer.
