Solving exponential equations with variable exponents.

Let’s solve each equation step by step.

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Problem 5: 3^(2x - 3) = 27

First, write 27 as a power of 3:
27 = 3³

So now:
3^(2x - 3) = 3³

Since the bases are the same, set the exponents equal:
2x - 3 = 3

Add 3 to both sides:
2x = 6

Divide by 2:
x = 3

✔ Check: Plug x = 3 back in → 3^(2*3 - 3) = 3^(6-3) = 3³ = 27 ✔️

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Problem 7: 6^(x + 1) = 36

Write 36 as a power of 6:
36 = 6²

So:
6^(x + 1) = 6²

Set exponents equal:
x + 1 = 2

Subtract 1 from both sides:
x = 1

✔ Check: 6^(1+1) = 6² = 36 ✔️

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Problem 9: 64^(x + 2) = 16^(x + 1)

This one needs a common base. Both 64 and 16 are powers of 4 (or 2). Let’s use base 4:

64 = 4³
16 = 4²

So rewrite:
(4³)^(x + 2) = (4²)^(x + 1)

Use exponent rule: (a^m)^n = a^(m*n)

Left side: 4^(3(x + 2)) = 4^(3x + 6)
Right side: 4^(2(x + 1)) = 4^(2x + 2)

Now set exponents equal:
3x + 6 = 2x + 2

Subtract 2x from both sides:
x + 6 = 2

Subtract 6:
x = -4

✔ Check:
Left: 64^(-4 + 2) = 64^(-2) = (4³)^(-2) = 4^(-6)
Right: 16^(-4 + 1) = 16^(-3) = (4²)^(-3) = 4^(-6) ✔️

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Final Answers:

5) x = 3
7) x = 1
9) x = -4

Final Answer:
5) x = 3
7) x = 1
9) x = -4