write as a single logarithm:

-2logbase3(1/x)+(1/3)logbase3(square root of x)

please show the steps to solving this.

thanx.

remember that

1 log (AxB) = log A + Log B (same base)
2 log (A/B) = log A - log B
3 log A^n = n log A

use these three rules and it becomes an easy question.

To write the expression -2log₃(1/x) + (1/3)log₃(√x) as a single logarithm, we can apply the rules of logarithms. Let's break it down step by step:

Step 1: Use the power rule of logarithms to simplify the square root in the second term:
log₃(√x) = log₃(x^(1/2)) = (1/2)log₃(x)

The expression now becomes:
-2log₃(1/x) + (1/3)(1/2)log₃(x)

Step 2: Apply the quotient rule of logarithms to simplify the first term:
log₃(1/x) = log₃(1) - log₃(x) = -log₃(x)

The expression now becomes:
-2(-log₃(x)) + (1/3)(1/2)log₃(x)

Step 3: Simplify the expression by distributing the coefficients and applying the addition rule of logarithms:
2log₃(x) + (1/6)log₃(x)

Step 4: Combine like terms:
(2 + 1/6)log₃(x) = (12/6 + 1/6)log₃(x) = (13/6)log₃(x)

Finally, the expression -2log₃(1/x) + (1/3)log₃(√x) can be written as a single logarithm:
(13/6)log₃(x)

So, the answer is (13/6)log₃(x).