Write as a Single Logarithm 2( log of 2x+ log of y)-( log of 3+2 log of 5)

Solution:

Step 1: Simplify each logarithmic term.

Step 1.1: Apply the product property of logarithms.

Using the product property $\log_b(x) + \log_b(y) = \log_b(xy)$, combine the logs:

$2 \log(2x) + 2 \log(y) - (\log(3) + 2 \log(5))$

Step 1.2: Use the coefficient rule to move the 2 inside the log.

Transform $2 \log(2xy)$ into $\log((2xy)^2)$:

$\log((2xy)^2) - (\log(3) + 2 \log(5))$

Step 1.3: Distribute the square over the product.

Apply the rule $(ab)^n = a^n b^n$:

$\log(2^2x^2y^2) - (\log(3) + 2 \log(5))$

Step 1.4: Calculate the square of 2.

Compute $2^2$:

$\log(4x^2y^2) - (\log(3) + 2 \log(5))$

Step 1.5: Simplify the remaining terms.

Step 1.5.1: Move the 2 inside the log for $\log(5)$.

Convert $2 \log(5)$ to $\log(5^2)$:

$\log(4x^2y^2) - (\log(3) + \log(25))$

Step 1.5.2: Calculate the square of 5.

Compute $5^2$:

$\log(4x^2y^2) - (\log(3) + \log(25))$

Step 1.6: Combine the logs with addition into a single log.

Use the product property again:

$\log(4x^2y^2) - \log(3 \cdot 25)$

Step 1.7: Multiply 3 by 25.

Calculate $3 \cdot 25$:

$\log(4x^2y^2) - \log(75)$

Step 2: Use the quotient property of logarithms.

Apply the quotient property $\log_b(x) - \log_b(y) = \log_b(\frac{x}{y})$ to combine the logs into a single logarithm:

$\log\left(\frac{4x^2y^2}{75}\right)$

Knowledge Notes:

1. Product Property of Logarithms: For any positive numbers $x$ and $y$ and any positive base $b$ (where $b \neq 1$), the logarithm of a product is the sum of the logarithms: $\log_b(xy) = \log_b(x) + \log_b(y)$.

2. Quotient Property of Logarithms: For any positive numbers $x$ and $y$ and any positive base $b$ (where $b \neq 1$), the logarithm of a quotient is the difference of the logarithms: $\log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y)$.

3. Power Rule of Logarithms: For any positive number $x$, any positive base $b$ (where $b \neq 1$), and any real number $n$, the logarithm of a power is the product of the exponent and the logarithm: $\log_b(x^n) = n \log_b(x)$.

4. Exponent Rules: The power of a product rule states that $(ab)^n = a^n b^n$, and the power of a power rule states that $(a^n)^m = a^{nm}$.

5. Combining Logarithms: When simplifying logarithmic expressions, it is often useful to combine multiple logarithms into a single logarithm using the product, quotient, and power rules. This is particularly helpful when solving equations involving logarithms or when trying to express a logarithmic relationship in a simpler form.