Problem 32 - Algebra 2

Question

Solve for \(n\) \(\log _{b} 8+3 \log _{b} n=3 \log _{b}(x-1)\)

Step-by-Step Solution

Answer

\(n = \frac{x-1}{2}\)

Step 1: Use the logarithm multiplication rule

Recall that the logarithm multiplication rule states \( a \log_b x = \log_b(x^a) \). Apply this rule to the expression \( 3 \log_b n \), rewriting it as \( \log_b(n^3) \). The equation now becomes:\[\log_b 8 + \log_b(n^3) = 3\log_b(x-1)\]

Step 2: Use the logarithm addition rule

The logarithm addition rule states that \( \log_b a + \log_b c = \log_b(ac) \). Apply this to the left side of the equation:\[\log_b(8 \cdot n^3) = 3\log_b(x-1)\]

Step 3: Handle the right side using the multiplication rule

Use the multiplication rule for logarithms again, rewriting the right side \( 3 \log_b(x-1) \) as:\[\log_b((x-1)^3)\]

Step 4: Equate the arguments of the logarithms

With the equation now being \( \log_b(8n^3) = \log_b((x-1)^3) \), set the arguments of the logarithms equal to each other:\[8n^3 = (x-1)^3\]

Step 5: Solve for \( n \)

To isolate \( n \), divide both sides by 8:\[n^3 = \frac{(x-1)^3}{8}\]Take the cube root of both sides:\[n = \sqrt[3]{\frac{(x-1)^3}{8}} = \frac{x-1}{2}\]

Key Concepts

Logarithm Multiplication Rule

The logarithm multiplication rule is a handy tool when dealing with logarithmic equations. This rule states that multiplying a logarithm by a constant is the same as raising the logarithm's argument to the power of that constant. In mathematical terms, it's expressed as:\[ a \log_b x = \log_b(x^a) \]For example, if you have \(3 \log_b n\), you can transform it to \(\log_b(n^3)\). This rule simplifies logarithmic expressions and helps in solving equations. When you're given a logarithmic term with a coefficient, consider using this rule to see if simplifying the expression might be helpful. This conversion often sets the stage for further simplifications using additional logarithmic properties.

Logarithm Addition Rule

The logarithm addition rule helps us combine two logarithms with the same base into a single logarithmic expression. It states that adding two logarithms is equivalent to the logarithm of the product of their arguments:\[ \log_b a + \log_b c = \log_b(ac) \]Imagine you have \(\log_b 8 + \log_b(n^3)\), this can be simplified using this rule to \(\log_b(8 \cdot n^3)\). This combined form is often easier to handle, especially when comparing or equating expressions on both sides of an equation. This can lead to quicker insights into solving or simplifying equations that involve multiple logarithmic terms.

Exponents

Exponents are a fundamental concept in mathematics, representing repeated multiplication. The exponent indicates how many times a base is multiplied by itself. For example, \(n^3\) means \(n \times n \times n\). Exponents come into play prominently within logarithmic functions, often due to the multiplication rule where expressions like \(x^a\) emerge from converting terms. When solving equations that involve logarithms, understanding how exponents work helps in translating between exponential and logarithmic forms. This translation is at the heart of steps like isolating variables in logarithmic equations.

Cube Root

The cube root is a special type of root that asks the question: "what number, when multiplied by itself twice, gives the original number?" It's denoted as \(\sqrt[3]{x}\) and is useful in solving equations involving cubes, like \(n^3\). When you encounter an equation such as \(n^3 = \frac{(x-1)^3}{8}\), taking the cube root on both sides simplifies the expression to find \(n\) directly. In this case:\[ n = \sqrt[3]{\frac{(x-1)^3}{8}} = \frac{x-1}{2} \]Understanding cube roots is essential in breaking down these expressions to solve equations, especially those resulting from using logarithm multiplication rules.