Problem 55 - Algebra and Trigonometry

Question

Expand: $$ \log _{b}\left(x^{3} \sqrt{y}\right) $$

Step-by-Step Solution

Answer

The expanded form of the expression is \(3 \log_b x + (1/2) \log_b y\)

Step 1: Apply Power Rule

Initially, apply the power rule to \(\log_b x^3\). This will change the expression to \(3 \log_b x \) . Similarly, we remember that square root is a power of (1/2), apply the power rule to \(\log_b y^{1/2}\) which will result to \((1/2) \log_b y\)

Step 2: Apply Product Rule

Now, apply the product rule which will therefore separate the two terms by a plus symbol: \(\log_b x^3 + \log_b y^{1/2}\). After applying the power rule in the first step the expression turns into \(3 \log_b x + (1/2) \log_b y\).

Step 3: Final Expression

After the application of the logarithmic rules, we arrived at the final result which is \(3 \log_b x + (1/2) \log_b y\). This is the expanded form of \(\log_b \left( x^3 \sqrt{y} \right)\). These could be further simplified based on the given value of variables x and y and base b; however, without additional information, this is the most simplified form.

Key Concepts

Power Rule in Logarithms

The power rule in logarithms is a fundamental property that simplifies expressions by handling exponents across multi-leveled operations. To understand this, you need to remember that whenever you see a logarithm of a power, like \( \log_b(x^n) \), you can simplify it by bringing the exponent down in front of the logarithm. This process effectively transforms the expression to \( n \log_b(x) \).

Essentially, the power rule allows us to convert a logarithmic expression that contains exponentiation into a simple multiplication of the exponent with the logarithm. This simplification makes it easier to deal with complex logarithmic expressions.

• Example: \( \log_b(x^3) = 3 \log_b(x) \).
• Root operations can be converted using the power rule: \( \sqrt{y} = y^{1/2} \) becomes \( (1/2) \log_b(y) \).

By applying this rule, you can begin to break down and simplify challenging logarithmic expressions with ease.

Product Rule in Logarithms

Logarithmic expressions often include products, and the product rule helps us deal with them efficiently. This rule states that the logarithm of a product is equal to the sum of the logarithms of its factors. Mathematically, it is expressed as \( \log_b(M \times N) = \log_b M + \log_b N \).

The product rule enables breaking down complicated expressions by splitting them into more manageable parts. When you apply the product rule to a logarithm with a product, you essentially transform it into an easier additive form.

• Example: \( \log_b(x^3 \sqrt{y})\).
• Applied: \( \log_b(x^3) + \log_b(\sqrt{y}) \).

This approach shows that it’s much simpler to handle multiple factors within a logarithmic expression when they are dealt with separately.

Expanding Logarithms

Expanding logarithms is about transforming a complex logarithmic expression into a simpler, more comprehensive form. This involves applying the power rule and the product rule sequentially. Expanding logarithms helps because it expresses the single logarithmic statement into several smaller, individual parts, making calculation and comprehension more straightforward.

In our case, the expression \( \log_b(x^3 \sqrt{y}) \) is expanded by first dealing with any powers or roots and then separating products.

• Initial expression: \( \log_b(x^3 \sqrt{y}) \).
• Apply the power rule to each: \( 3\log_b x + (1/2)\log_b y \).

By expanding logarithms, you translate a complex equation into a simpler one, ensuring that each part is clearer and more approachable. This approach can be especially useful in calculus and other advanced math classes, where complex expressions are commonplace.