Problem 7 - Precalculus

Question

Expand the given logarithm and simplify. Assume when necessary that all quantities represent positive real numbers. $$ \log _{\sqrt{2}}\left(4 x^{3}\right) $$

Step-by-Step Solution

Answer

The expanded and simplified form is \(2 + \frac{6 \log(x)}{\log(2)}\).

Step 1: Apply the Change of Base Formula

To expand the logarithm \( \log_{\sqrt{2}}(4x^3) \), we first use the change of base formula. This converts the logarithm with base \(\sqrt{2}\) into a fraction using base 10 or base \(e\). \[ \log_{\sqrt{2}}(4x^3) = \frac{\log(4x^3)}{\log(\sqrt{2})} \]

Step 2: Simplify the Base

Since \( \sqrt{2} = 2^{1/2} \), we can rewrite \( \log(\sqrt{2}) \). Using the property \( \log(a^b) = b \log(a) \), we have: \[ \log(\sqrt{2}) = \frac{1}{2} \log(2) \]

Step 3: Expand the Numerator

Expand \( \log(4x^3) \) using the property of logarithms for multiplication, \( \log(ab) = \log(a) + \log(b) \). Thus: \[ \log(4x^3) = \log(4) + \log(x^3) \]

Step 4: Simplify \(\log(x^3)\)

Use the power rule for logarithms \( \log(a^b) = b \log(a) \) to simplify \( \log(x^3) \): \[ \log(x^3) = 3 \log(x) \]

Step 5: Substitute and Simplify the Expression

Substitute \( \log(4) + \log(x^3) \) in the numerator and \( \frac{1}{2} \log(2) \) in the denominator. Then, simplify the expression: \[ \frac{\log(4) + 3\log(x)}{\frac{1}{2} \log(2)} \] Which simplifies further by multiplying numerator and denominator by 2: \[ \frac{2(\log(4) + 3 \log(x))}{\log(2)} \] Divide both terms by \( \log(2) \): \[ \frac{2 \log(4)}{\log(2)} + \frac{6 \log(x)}{\log(2)} \]

Step 6: Evaluate \(\frac{\log(4)}{\log(2)}\)

Using the property that \( \log_b(a^n) = n \log_b(a) \), evaluate \( \frac{\log(4)}{\log(2)} \). Since \( 4 = 2^2 \), it follows: \[ \log(4) = 2 \log(2) \] Hence, \( \frac{2 \log(2)}{\log(2)} = 2 \).

Step 7: Write the Final Expression

Substituting \(\frac{2 \log(4)}{\log(2)} = 2\) back into the expression, we get: \[ 2 + \frac{6 \log(x)}{\log(2)} \]

Key Concepts

Change of Base Formula

The change of base formula is a handy tool that allows us to convert a logarithm with any base into a form using either base 10 or base \(e\), which is the natural logarithm. This is useful because we often have calculators or software that can compute logarithms with these two bases easily. To apply this, if you have a logarithm \(\log_b(a)\), the change of base formula is:

• \(\log_b(a) = \frac{\log_c(a)}{\log_c(b)}\)

Where \(c\) can be any base, but we typically use 10 or \(e\). In our example, converting \(\log_{\sqrt{2}}(4x^3)\) means using this formula to rewrite it as \(\frac{\log(4x^3)}{\log(\sqrt{2})}\), which simplifies further using logarithm properties.

Logarithm Properties

Understanding logarithmic properties is crucial for both expanding and simplifying logarithmic expressions. Here are some key properties:

• Product Rule: \( \log(a \cdot b) = \log(a) + \log(b) \)
• Power Rule: \( \log(a^b) = b \cdot \log(a) \)
• Quotient Rule: \( \log\left( \frac{a}{b} \right) = \log(a) - \log(b) \)

These properties allow us to break down complex logarithmic expressions into simpler terms. For instance, in our example, \(\log(4x^3)\) expands to \(\log(4) + \log(x^3)\), and using the power rule, \(\log(x^3)\) turns into \(3 \log(x)\). This makes understanding and manipulating logarithms much more manageable.

Logarithm Expansion

Logarithm expansion involves using logarithmic properties to break down a single logarithmic expression into a sum or difference of multiple terms. This process makes solving logarithmic equations and performing computations simpler. It begins with identifying components that can be expanded using these properties. For example, let's expand \(\log(4x^3)\):

• First, apply the product rule: \(\log(4x^3) = \log(4) + \log(x^3)\)
• Next, use the power rule on \(\log(x^3)\), giving us \(3 \log(x)\)

Combining these steps provides us with an expanded form of the logarithm, \(\log(4) + 3 \log(x)\), which can be further simplified by substitution and division in the change of base formula.

Simplifying Logarithmic Expressions

Simplifying logarithmic expressions is about making them as concise and clear as possible while ensuring they are accurate and equivalent to their original form. This often involves using known values and properties to reduce terms. In our exercise:

• First, rewrite logarithms using consistent bases if possible. For example, simplify \(\log(\sqrt{2})\) to \(\frac{1}{2} \log(2)\) because we know \( \sqrt{2} = 2^{1/2} \).
• Then use operations like factoring and cancellation to simplify the whole expression. Multiply through if necessary to remove fractions in the form: \(\frac{2(\log(4) + 3 \log(x))}{\log(2)}\).
• Finally, evaluate expressions by substituting known logarithmic values, such as \(\frac{\log(4)}{\log(2)} = 2\), leading to a simplified expression like \(2 + \frac{6 \log(x)}{\log(2)}\).

This process ensures that we arrive at a simpler expression without altering its value.