Problem 11 - Algebra 2

Question

Write each logarithmic expression as a single logarithm. \(\log 7+\log 2\)

Step-by-Step Solution

Answer

The simplified form of the expression \(\log 7 + \log 2\) is \(\log 14\).

Step 1: Identifying the Property of Logarithms

To solve this, it is important to recognize that the given equation can be re-written using the property of logarithms. Specifically, the rule that concerns us in this case is \(\log_b(x) + \log_b(y) = \log_b(xy)\).

Step 2: Applying the Property of Logarithms

Let's now apply the aforementioned rule to the given expression which is \(\log 7 + \log 2\). This will transform our expression into a single logarithm. Therefore, \(\log 7 + \log 2\) becomes \(\log(7 * 2)\).

Step 3: Simplifying the Result

Our final step is simple: we just need to simplify the result in the brackets. So, \(\log(7 * 2)\) simplifies to \(\log 14\).

Key Concepts

logarithmic expressions

Logarithmic expressions are a way of writing numbers that tells you what power you must raise a base number to, in order to get another number. Logarithms are often used to simplify calculations with very large or very small numbers by converting multiplication into addition. For example, in the expression \( \log 7 + \log 2 \), we are dealing with two logarithmic expressions. Each of these expressions represents an exponent to which the base, often 10, must be raised to reach 7 and 2, respectively. Understanding how to manipulate these expressions can make complex problems more manageable. By translating multiplication, division, and other operations into simpler addition and subtraction problems, we can solve these expressions more easily.

properties of logarithms

Properties of logarithms help in transforming and simplifying logarithmic expressions. One of the primary properties relevant to this problem is the product rule. The product rule for logarithms states that the sum of two logarithms is the same as the logarithm of the product of their arguments: \( \log_b(x) + \log_b(y) = \log_b(xy) \). This property allows us to combine logarithmic terms, making expressions simpler to work with. In the exercise given, the expression \( \log 7 + \log 2 \) can be rewritten as \( \log(7 \times 2) \) using this rule. These properties are essential tools in the toolkit of anyone working with logarithmic functions, making it easier to manipulate and interpret these expressions in algebra and calculus.

simplifying logarithms

Simplifying logarithms involves combining multiple logarithmic terms into a single term whenever possible. This process often uses properties of logarithms like the product, quotient, and power rules. In our example, \( \log 7 + \log 2 \) is simplified by using the product rule, resulting in \( \log 14 \). Simplification makes the expressions more concise and easier to work with, especially in algebraic equations where multiple logarithms are present. Often, students initially find these tasks daunting, but with practice and understanding of the properties, simplification becomes an intuitive part of solving logarithmic equations.

Algebra 2

In the Algebra 2 curriculum, logarithms are an important concept that students learn to understand and manipulate. They provide a foundation for solving exponential equations and are a stepping stone to more advanced topics. During Algebra 2, students encounter problems like the exercise given, which involves applying properties of logarithms to write expressions in a simpler form. The importance lies in recognizing patterns and applying rules correctly to simplify expressions and solve equations. Mastering these skills prepares students for calculus and other higher math courses where logarithms continue to play a crucial role.