To solve the first equation, we need to simplify it:
7^x+2 + 27^x-1 = 377^x 7^2 + 2 (7^x / 7) = 377^x 49 + 2 (7^x / 7) = 3749 7^x + 2 (7^x / 7) = 3749 7^x + 14/7 7^x = 3749 7^x + 14 7^x = 3763 7^x = 377^x = 375 / 67^x = 5
Now, we need to solve for x using logarithms:
log7(7^x) = log7(5x = log7(5)
For the second equation, simplify it first:
2^(2x-9) < 2^(2x) 2^(-9) < 2^(2x) (1/2^9) < 2^(2x) * (1/512) < 2^(2x) < 512
Now, 512 = 2^9, so the inequality becomes:
2^(2x) < 2^2x < x < 4.5
Therefore, the solution to the second equation is x < 4.5.
For the third equation, we have:
log3(27) - log9(81log3(3^3) - log9(9^23log3(3) - 2log9(331 - 2(1/23 - 2
Therefore, log3(27) - log9(81) = 2.
To solve the first equation, we need to simplify it:
7^x+2 + 27^x-1 = 37
7^x 7^2 + 2 (7^x / 7) = 37
7^x 49 + 2 (7^x / 7) = 37
49 7^x + 2 (7^x / 7) = 37
49 7^x + 14/7 7^x = 37
49 7^x + 14 7^x = 37
63 7^x = 37
7^x = 375 / 6
7^x = 5
Now, we need to solve for x using logarithms:
log7(7^x) = log7(5
x = log7(5)
For the second equation, simplify it first:
2^(2x-9) <
2^(2x) 2^(-9) <
2^(2x) (1/2^9) <
2^(2x) * (1/512) <
2^(2x) < 512
Now, 512 = 2^9, so the inequality becomes:
2^(2x) < 2^
2x <
x < 4.5
Therefore, the solution to the second equation is x < 4.5.
For the third equation, we have:
log3(27) - log9(81
log3(3^3) - log9(9^2
3log3(3) - 2log9(3
31 - 2(1/2
3 -
2
Therefore, log3(27) - log9(81) = 2.