Logarithm Problem Solved: Finding log 6

To find the value of log 6 when log 2 = x and log 3 = y are given, we need to use a fundamental property of logarithms. This type of problem often appears in basic algebra and pre-calculus courses, testing your understanding of logarithm rules.

Logarithm Properties Explained

One of the essential properties of logarithms is the product rule. This rule states that the logarithm of a product of two numbers is equal to the sum of the logarithms of those individual numbers. Mathematically, for any positive numbers \(M\) and \(N\), and a base \(b\) (where \(b > 0\) and \(b \ne 1\)), the product rule is expressed as:

\[ \log_b(MN) = \log_b M + \log_b N \]

In this specific problem, the base of the logarithm is not explicitly mentioned. When the base is not written, it commonly implies either a common logarithm (base 10) or a natural logarithm (base e). However, the product rule applies consistently regardless of the base used, as long as it remains the same throughout the calculation.

Solving for Log 6

Our goal is to find log 6. We know that the number 6 can be expressed as a product of its prime factors, 2 and 3:

\[ 6 = 2 \times 3 \]

Now, we can apply the logarithm product rule to this expression. Taking the logarithm of both sides of the equation \(6 = 2 \times 3\), we get:

\[ \log 6 = \log (2 \times 3) \]

Using the product rule of logarithms, we can expand the right side of the equation:

\[ \log (2 \times 3) = \log 2 + \log 3 \]

The problem provides us with specific values for log 2 and log 3:

• log 2 = x
• log 3 = y

By substituting these given values into our expanded logarithmic expression, we can find the value of log 6:

\[ \log 6 = x + y \]

Therefore, based on the fundamental properties of logarithms and the given expressions for log 2 and log 3, the value of log 6 is x + y.