Question

Write each expression as a single logarithm. \(8 \log _{2} \sqrt{3 x-2}-\log _{2}\left(\frac{4}{x}\right)+\log _{2} 4\)

Short answer

\text{log}_{2} \frac{(3x-2)^{4} \times x}{4}

Step-by-step solution

- Apply Power Rule

The power rule of logarithms states that \(k \times \text{log}_{b} M = \text{log}_{b} M^{k}\). Apply this rule to the term \(8 \text{log}_{2} \text{√}(3x-2)\). This gives us: \[8 \text{log}_{2} \text{√}(3x-2) = \text{log}_{2} (\text{√}(3x-2))^{8} = \text{log}_{2} (3x-2)^{4}\].

- Simplify Second Term

Recognize that \( \frac{4}{x} = 4x^{-1} \). Rewrite the logarithm: \[ - \text{log}_{2} \frac{4}{x} = - \text{log}_{2} (4x^{-1}) = - (\text{log}_{2} 4 + \text{log}_{2} x^{-1})\]. According to the power rule \( \text{log}_{2} x^{-1} = - \text{log}_{2} x \), the expression transforms to: \[ - (\text{log}_{2} 4 + (- \text{log}_{2} x)) = -\text{log}_{2} 4 + \text{log}_{2} x \].

- Combine Logarithms

Now, we have: \[ \text{log}_{2} (3x-2)^{4} + \text{log}_{2} x - \text{log}_{2} 4 \]. Using the product rule, \( \text{log}_{b} M + \text{log}_{b} N = \text{log}_{b} (M \times N)\), and the quotient rule, \( \text{log}_{b} M - \text{log}_{b} N = \text{log}_{b} \frac{M}{N}\), combine the terms: \[ \text{log}_{2} \frac{(3x-2)^{4} \times x}{4} \].

- Final Expression

The expression as a single logarithm is: \[ \text{log}_{2} \frac{(3x-2)^{4} \times x}{4} \].

Key concepts

Logarithm Rules

To solve logarithmic expressions like the one in the exercise, it's crucial to understand the fundamental rules of logarithms.
These rules help in rewriting and simplifying complex logarithmic terms.

• Product Rule: This states that \(\text{log}_{b}(MN) = \text{log}_{b}M + \text{log}_{b}N\).
• Quotient Rule: This indicates that \(\text{log}_{b}\frac{M}{N} = \text{log}_{b} M - \text{log}_{b} N\).
• Power Rule: According to this rule, \(\text{log}_{b}(M^{k}) = k \times \text{log}_{b} M\).
• Change of Base Formula: This helps in converting logarithms to a different base: \(\text{log}_{b} M = \frac{ \text{log}_{c} M}{ \text{log}_{c} b}\).

Understanding these rules can help you simplify and combine multiple logarithmic terms into a single, more manageable expression.

Power Rule

The Power Rule is essential when dealing with coefficients in front of logarithms.
Given by \(\text{log}_{b} M^{k} = k \times \text{log}_{b} M\), it’s the rule we'll apply to some terms in the exercise.
For example, in the term \(8 \text{log}_{2} \text{√}(3x-2)\), the coefficient 8 can be moved as an exponent of the argument: \[8 \text{log}_{2} \text{√}(3x-2) = \text{log}_{2} (\text{√}(3x-2))^{8} = \text{log}_{2} (3x-2)^{4} \]. This transformation simplifies the expression significantly and makes it easier to combine with other logarithmic terms.
Whenever you see a number multiplied in front of a logarithm, think about how the Power Rule can help reframe that term.

Product Rule

The Product Rule helps in combining logarithms that involve the product of two or more arguments.
The rule \(\text{log}_{b}(MN) = \text{log}_{b} M + \text{log}_{b} N\) allows us to add the logarithms of two numbers when their arguments are multiplied.
For instance, in ,...\ (3x-2)^{4} \ and \ \text {log}_{2} x- \ these can be combined using the Product Rule:
\ \text {log}_{2} ( (3x-2)^{4} \times x} \.
This combination simplifies the problem, transforming multiple logarithmic terms into a single, unified logarithmic expression.

Quotient Rule

The Quotient Rule is useful for simplifying differences of logarithms.
It states that \(\text{log}_{b}\frac{M}{N} = \text{log}_{b} M - \text{log}_{b} N\).
In the exercise provided, \(\text{log}_{2} ( (3x-2)^{4}) - \text{log}_{2} 4 \), this rule helps combine terms into a single logarithm:
\[ \text{log}_{2}\frac{(3x-2)^{4} \times x}{4} \].
Applying the Quotient Rule often helps to consolidate and simplify expressions, providing an effective way to solve complex logarithmic problems.