Problem 25 - Algebra A Combined Function

Question

Write each as a single logarithm. Assume that variables represent positive numbers. $$ 3 \log _{4} 2+\log _{4} 6 $$

Step-by-Step Solution

Answer

The expression simplifies to \( \log_{4} 48 \).

Step 1: Recall the Logarithm Power Rule

The power rule of logarithms states that \( a \cdot \log_b(x) = \log_b(x^a) \). This means we can move the coefficient 3 in front of \( \log_4(2) \) as an exponent within the logarithm. So, \( 3 \log_{4} 2 \) becomes \( \log_{4} (2^3) \).

Step 2: Simplify using the Power Rule

From Step 1, we have \( 3 \log_{4} 2 = \log_{4} (2^3) = \log_{4} 8 \). This simplifies the expression by using the exponentiation property.

Step 3: Use the Logarithm Product Rule

Now apply the product rule for logarithms, which states \( \log_b(x) + \log_b(y) = \log_b(xy) \). This allows us to combine the two logarithms \( \log_{4} 8 \) and \( \log_{4} 6 \) using multiplication: \( \log_{4} 8 + \log_{4} 6 = \log_{4} (8 \cdot 6) \).

Step 4: Compute the Product

Multiply the numbers inside the logarithm from Step 3: \( 8 \times 6 = 48 \). Thus, the expression \( \log_{4} (8 \cdot 6) \) is rewritten as \( \log_{4} 48 \).

Key Concepts

Logarithm Power Rule

The logarithm power rule is a handy tool when dealing with logarithms that involve coefficients in front of the log expression. This rule states that if you have a coefficient multiplied by a logarithm, you can move this coefficient as an exponent inside the logarithm. For example, if you have \(a \cdot \log_b(x)\), you can rewrite it as \(\log_b(x^a)\). This makes it easier to simplify and manipulate more complex logarithmic expressions. In our example, the coefficient of 3 in front of \( \log_4(2) \) becomes an exponent: \(3 \log_4(2) = \log_4(2^3)\). After simplification, \(2^3\) becomes 8, leading to \(\log_4(8)\). With this rule, you effectively transform multiplicative expressions into simpler exponential forms.

Logarithm Product Rule

The logarithm product rule is another essential concept that helps combine and simplify logarithmic expressions. This rule allows you to add logarithms together if they share the same base, converting the sum into a single logarithm by multiplying their arguments. Mathematically, this rule is expressed as \(\log_b(x) + \log_b(y) = \log_b(xy)\). By using this rule, you can merge multiple small logarithmic expressions into one larger one. Join \(\log_4(8)\) and \(\log_4(6)\) together: \(\log_4(8) + \log_4(6) = \log_4(8 \cdot 6)\). This not only simplifies the form but also makes calculations involving logarithms more streamlined. Apply this logic effectively to deal with more intricate expressions efficiently.

Simplifying Expressions

Simplifying expressions is the process of making mathematical expressions easier to understand or work with, usually by reducing them to a form that is simpler to handle. When simplifying logarithmic expressions, we often use rules like the power rule and product rule to consolidate multiple terms into a single, cohesive expression. In our exercise, we started with the expression \(3 \log_4 2 + \log_4 6\). By systematically applying the power rule to adjust the coefficient and the product rule to combine terms, we achieve the simplified expression \(\log_4 48\). With these steps, we've taken a seemingly complex expression and reduced it to its simplest form. This process not only aids in solving problems more efficiently but also enhances comprehension of underlying mathematical principles.