The exponential equation can be written as:
3 (4 y + 2) = 1
9 (y - 2) 27
Step 1: Express everything with base 3
We know:
9 = 32
27 = 33
NB - 1/27 becomes 3 - 3
So rewrite:
3 (4 y + 2) = 3 - 3
(3 2) y - 2
Step 2: Apply exponent rules
(32) y-2 = 3 2(y -2) = 3 2y - 4
Now the equation becomes:
3 4y + 2 = 3 -3
3 2y - 4
Step 3: Use the quotient rule for exponents
3 4y + 2
3 2y - 4
= 3 (4y + 2) - (2y - 4)
= 3 (4y + 2 - 2y + 4)
= 3 2y + 6
3 2y + 6 = 3 - 3
Step 4: Equating exponents (since bases are the same)
2y + 6 = −3
2y = −9
y = − 9/2
Final Answer:
y = − 9/2
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