Question

Write each expression as a sum and/or difference of logarithms. Express powers as factors. $$ \log \left[\frac{x(x+2)}{(x+3)^{2}}\right] \quad x>0 $$

Short answer

\[ \log x + \log(x + 2) - 2 \log(x + 3) \]

Step-by-step solution

Apply the Quotient Rule

Use the quotient rule for logarithms, which states that \( \log \left(\frac{A}{B}\right) = \log A - \log B \). Here, let \ A = x(x + 2) \ and \ B = (x + 3)^2 \. Apply the quotient rule: \[ \log \left( \frac{x(x + 2)}{(x + 3)^2}\right) = \log(x(x + 2)) - \log( (x + 3)^2 ) \]

Apply the Product Rule

Next, use the product rule for logarithms, which states that \( \log(AB) = \log A + \log B \). Apply it to \ \log(x(x + 2)) \ to get: \[ \log(x(x + 2)) = \log x + \log(x + 2) \]

Simplify the Exponent

Use the power rule for logarithms on \ \[ \log((x + 3)^2) = 2 \log(x + 3) \]. Substitute this expression back into the equation from Step 1, obtaining: \[ \log(x(x + 2)) - \log((x + 3)^2) = \log x + \log(x + 2) - 2 \log(x + 3) \]

Key concepts

Quotient Rule for Logarithms

The Quotient Rule for logarithms is a vital concept. It helps simplify expressions involving division inside a logarithm. Specifically, it states: \( \log \left( \frac{A}{B} \right) = \log A - \log B \).

In simpler terms:

• If you have a logarithm of a fraction, you can break it into the subtraction of two separate logarithms.

In our exercise:
\( \log \left[\frac{x(x+2)}{(x+3)^{2}}\right] \)

Here, think of the entire numerator as one part, and the entire denominator as another part.
So letting \( A = x(x + 2) \) and \( B = (x + 3)^2 \),

We get to: \[ \log \left( \frac{x(x + 2)}{(x + 3)^2} \right) = \log(x(x + 2)) - \log((x + 3)^2) \]

Product Rule for Logarithms

The Product Rule for logarithms simplifies expressions involving multiplication inside a logarithm. It states: \( \log(AB) = \log A + \log B \).

Essentially:

• If you have a logarithm of a product, you can split it into the sum of two separate logarithms.

In our example:
\[ \log(x(x + 2)) = \log x + \log(x + 2) \]

Here:

• We treat \( x \) and \( x + 2 \) as separate parts inside the multiplication.

Hence, \( \log(x(x + 2)) \) breaks down into \( \log x + \log(x + 2) \)

Power Rule for Logarithms

The Power Rule for logarithms helps when you have an exponent inside a logarithm. This rule states: \( \log(A^k) = k \log A \).

Simply put:

• If you have a logarithm of a number raised to a power, you can bring the exponent in front of the logarithm as a multiplier.

In our exercise:
\[ \log((x + 3)^2) = 2 \log(x + 3) \]

Here, we treat the exponent \( 2 \) as a multiplier outside the logarithm.
This helps simplify our final expression into:
\[ \log(x(x + 2)) - \log((x + 3)^2) = \log x + \log(x + 2) - 2 \log(x + 3) \]

Using these rules, we've broken the original complex logarithmic expression into simpler parts!