Simplify: log3^27+2log3^9-log3^54
To simplify the expression, we can use the logarithmic rules.
First, let's write each term using the logarithmic rule for exponentiation:
log3^27 = 3log3(27)
log3^9 = 9log3(9)
log3^54 = 54log3(3)
Now we can rewrite the expression:
3log3(27) + 2(9log3(9)) - 54log3(3)
Using the logarithmic rule for multiplication, we can simplify further:
= 3log3(27) + 18log3(9) - 54log3(3)
Since 27 is equal to 3^3 and 9 is equal to 3^2, we can simplify the logarithms:
= 3log3(3^3) + 18log3(3^2) - 54log3(3)
Using the logarithmic rule for simplifying exponents within logarithms, we get:
= 3(3) + 18(2) - 54(1)
= 9 + 36 - 54
= -9
Therefore, the simplified expression is -9.