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Solve for [tex]\( x \)[/tex]:

[tex]\[ 7^{2x+3} = 343^x \][/tex]

Sagot :

GinnyAnsweridn
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.
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