Problem 47 - Algebra and Trigonometry

Question

Use the Laws of Logarithms to combine the expression. $$ \frac{1}{3} \log (2 x+1)+\frac{1}{2}\left[\log (x-4)-\log \left(x^{4}-x^{2}-1\right)\right] $$

Step-by-Step Solution

Answer

\( \log \left(\frac{(2x+1)^{1/3} \cdot (x-4)^{1/2}}{(x^{4} - x^{2} - 1)^{1/2}}\right) \)

Step 1: Apply the Power Rule

The given expression is \( \frac{1}{3} \log (2x+1) + \frac{1}{2} [\log (x-4) - \log (x^{4} - x^{2} -1)] \). We start by applying the Power Rule, \( a \log b = \log b^{a} \), to get \( \log((2x+1)^{1/3}) + \log((x-4)^{1/2}) - \log((x^{4} - x^{2} - 1)^{1/2}) \).

Step 2: Apply the Product Rule

Use the Product Rule, \( \log a + \log b = \log(ab) \), to combine the logarithms: \( \log((2x+1)^{1/3} \cdot (x-4)^{1/2}) \). Now the expression is \( \log(((2x+1)^{1/3} \cdot (x-4)^{1/2})) - \log((x^{4} - x^{2} - 1)^{1/2}) \).

Step 3: Apply the Quotient Rule

Now, apply the Quotient Rule, \( \log a - \log b = \log \left(\frac{a}{b}\right) \), to combine the remaining expression into one logarithm: \( \log \left(\frac{(2x+1)^{1/3} \cdot (x-4)^{1/2}}{(x^{4} - x^{2} - 1)^{1/2}}\right) \).

Step 4: Simplify the Final Expression

The expression \( \log \left(\frac{(2x+1)^{1/3} \cdot (x-4)^{1/2}}{(x^{4} - x^{2} - 1)^{1/2}}\right) \) is the combined form of the original problem using the Laws of Logarithms. This is the simplest expression possible.

Key Concepts

Power Rule

Understanding the Power Rule is essential when working with logarithmic expressions. The Power Rule allows you to move a constant multiplier from in front of a logarithm to the exponent of the logarithmic argument. In formula terms, this rule is expressed as:

• \( a \log b = \log b^{a} \)

This means that any number multiplied by a logarithm can be turned into an exponent. For example, if you have an expression \( \frac{1}{3} \log (2x+1) \), you use the Power Rule to rewrite it as \( \log((2x+1)^{1/3}) \).
The same applies to the term \( \frac{1}{2} \log (x-4) \), changing it to \( \log((x-4)^{1/2}) \). This step is crucial because it simplifies expressions and prepares them for further combination using other rules. Remember, understanding exponents as repeated multiplication, the Power Rule often simplifies complex expressions before proceeding to the next steps.

Product Rule

The Product Rule is used when you want to combine two logarithms that are added together. With logarithms, addition inside the log is equivalent to multiplication outside it. The formula for the Product Rule is:

• \( \log a + \log b = \log(ab) \)

This rule simplifies expressions by merging separate logarithmic terms into one. For instance, when you have terms like \( \log((2x+1)^{1/3}) + \log((x-4)^{1/2}) \), the Product Rule lets you combine these into \( \log(((2x+1)^{1/3} \cdot (x-4)^{1/2})) \).
By converting the addition of logs into a multiplication inside a single log, we reduce the complexity of the expression. The result is a more compact form that's easier to manage and calculate. It's an elegant way to reshape your logarithmic expressions to be as concise as possible before implementing further laws, such as the Quotient Rule.

Quotient Rule

The Quotient Rule simplifies expressions with logarithms that are subtracted, just like division inside a single logarithm. The rule can be formulated as follows:

• \( \log a - \log b = \log \left(\frac{a}{b}\right) \)

This is particularly useful when dealing with differences between two logarithmic terms. Take the expression \( \log(((2x+1)^{1/3} \cdot (x-4)^{1/2})) - \log((x^{4} - x^{2} - 1)^{1/2}) \). By applying the Quotient Rule, you can combine them into one: \( \log \left(\frac{(2x+1)^{1/3} \cdot (x-4)^{1/2}}{(x^{4} - x^{2} - 1)^{1/2}}\right) \).
This often reveals the underlying relationships between the factors involved. Not only does it lead to a neater expression, but it also provides clarity, helping to see the overall structure more clearly. By compressing several logs into a single logarithmic function, the expression is simplified, making it easier to evaluate or manipulate further.