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 $(ab{)}^n=a^n{b}^n$:

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

Step 1.4: Calculate the square of 2.

Compute $2^2$:

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

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 $\log (5^2)$:

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

Step 1.5.2: Calculate the square of 5.

Compute $5^2$:

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

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

Use the product property again:

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

Step 1.7: Multiply 3 by 25.

Calculate $3\cdot 25$:

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

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:

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

Knowledge Notes:

1. Product Property of Logarithms: For any positive numbers $x$ and $y$ and any positive base $b$ (where $b\ne 1$), 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 $b\ne 1$), 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 $b\ne 1$), 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.