Write the logarithm of a single quantity: 2ln(x)+ln(4)-(1/2)ln(9)

Think on these:

n ln x = ln (x^n)

lnx + lny= ln (xy)

lnx-lny= ln(x/y)

To find the logarithm of the expression 2ln(x) + ln(4) - (1/2)ln(9), we need to remember the properties of logarithms.

The first property is the logarithm of a product:
log(a * b) = log(a) + log(b)

The second property is the logarithm of a quotient:
log(a / b) = log(a) - log(b)

The third property is the logarithm of an exponent:
log(a^b) = b * log(a)

Using these properties, we can simplify the given expression step by step:

1. Apply the property of logarithm of an exponent to the first term:
2ln(x) = ln(x^2)

2. Combine the logarithms of the first and second terms using the property of logarithm of a product:
ln(x^2) + ln(4) = ln(x^2 * 4) = ln(4x^2)

3. Apply the property of logarithm of a quotient to the third term:
(1/2)ln(9) = ln(9^(1/2)) = ln(√9) = ln(3)

Now, the expression becomes:
ln(4x^2) - ln(3)

4. Apply the property of logarithm of a quotient to simplify the expression:
ln(4x^2 / 3)

Therefore, the logarithm of the given expression is ln(4x^2 / 3).