Question

Write each expression as a sum and/or difference of logarithms. Express powers as factors. $$ \log \left[\frac{x^{3} \sqrt{x+1}}{(x-2)^{2}}\right] \quad x>2 $$

Short answer

3 \log x + 1/2 \log (x+1) - 2 \log (x-2)

Step-by-step solution

Apply the Quotient Rule of Logarithms

Use the property \(\log \frac{A}{B} = \log A - \log B \)to split the given logarithm into two separate logarithms: \(\log \left[ \frac{x^{3} \sqrt{x+1}}{(x-2)^{2}} \right] \) becomes \(\log (x^{3} \sqrt{x+1}) - \log ( (x-2)^{2} ) \).

Apply the Product Rule of Logarithms

Use the property \(\log (AB) = \log A + \log B \) to further break down \(\log (x^{3} \sqrt{x+1}) \).So, \(\log (x^{3} \sqrt{x+1}) \) becomes \(\log x^{3} + \log \sqrt{x+1} \).

Simplify Using the Power Rule of Logarithms

Use the property \(\log A^{B} = B \log A \) to simplify each term separately:1. For \(\log x^{3} \), apply \(3 \log x \).2. For \(\log \sqrt{x+1} \), remember that \(\sqrt{x+1} = (x+1)^{1/2} \), so apply \(\frac{1}{2} \log (x+1) \).3. For \(\log ( (x-2)^{2} ) \), apply \(2 \log (x-2) \).Therefore, \(\log (x^{3}) = 3 \log x \), \(\log \sqrt{x+1} = \frac{1}{2} \log (x+1) \), and \(\log (x-2)^{2} = 2 \log (x-2) \).

Combine All Terms

Combine all simplified terms to write the expression as a sum or difference of logarithms. The final expression is:\[3 \log x + \frac{1}{2} \log (x+1) - 2 \log (x-2) \].

Key concepts

Quotient Rule of Logarithms

The quotient rule of logarithms is a powerful property that allows you to break down complex logarithmic expressions. The rule states that the logarithm of a quotient is equal to the difference between the logarithms of the numerator and the denominator.
The formula for the quotient rule is: \ \( \log \left( \frac{A}{B} \right) = \log A - \log B \).
For example, if you have \( \log \left( \frac{16}{4} \right) \), you can simplify it as \( \log 16 - \log 4 \).
This rule is especially useful when simplifying expressions like \( \log \left( \frac{x^{3} \sqrt{x+1}}{(x-2)^{2}} \right) \). By applying the quotient rule, we can split this into:
\( \log (x^{3} \sqrt{x+1}) - \log ((x-2)^{2}) \).
This step simplifies the problem and makes it easier to handle.

Product Rule of Logarithms

The product rule of logarithms helps simplify expressions that involve the logarithm of a product by breaking them into a sum of individual logarithms.
The formula for the product rule is: \( \log (AB) = \log A + \log B \).
For instance, if you have \( \log (3 \cdot 2) \), you can rewrite it as \( \log 3 + \log 2 \).
In the context of our example: \( \log (x^{3} \sqrt{x+1}) \), we can simplify it into \( \log x^{3} + \log \sqrt{x+1} \).
This rule helps break down complex expressions into manageable parts before applying further simplifications. This step-by-step breakdown ensures that you manage smaller pieces, making the entire problem easier to solve.

Power Rule of Logarithms

The power rule of logarithms allows you to handle exponents inside logarithms by pulling them out as factors.
The formula for the power rule is: \( \log A^{B} = B \log A \).
For example, \( \log (4^{3}) \) can be simplified to \( 3 \log 4 \).
In our exercise, consider the expression \( \log x^{3} \): it can be simplified to \( 3 \log x \).
Similarly, for \( \log \sqrt{x+1} \), we first rewrite it as \( \log (x+1)^{1/2} \). Applying the power rule, it becomes \( \frac{1}{2} \log (x+1) \).
Lastly, \( \log ((x-2)^{2}) \) becomes \( 2 \log (x-2) \).
Using the power rule, you can turn complicated logarithmic expressions into simpler ones, making the entire process more straightforward.