Simplify: log3^27+2log3^9-log3^54

Answer 1

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.