Problem 61 - Algebra 2

Question

Write true or false for each statement. Justify your answer. \(\log _{5} 16-\log 2=\log _{5} 8\)

Step-by-Step Solution

Answer

The statement is true. After simplifying the given expression, both sides of the equations are equal.

Step 1: Use log property

We can make use of the logarithm property \(\log a - \log b = \log \frac{a}{b}\) to simplify the left side of the equation: \(\log _{5} 16 - \log 2 = \log _{5} \frac{16}{2}\)

Step 2: Simplify the expression

Simplify the expression on the right side as: \(\log _{5} \frac{16}{2} = \log _{5} 8\)

Step 3: Compare both sides

Now, we compare the simplified expression with the right side i.e., \(\log _{5} 8 = \log _{5} 8\)

Key Concepts

Logarithm Properties

Logarithm properties are special rules that make it easier to work with logarithmic expressions. These properties are particularly handy when simplifying equations or solving logarithmic expressions. One of the fundamental properties is the logarithmic identity \(\log_a b - \log_a c = \log_a \frac{b}{c}\). This property tells us how to condense the difference of two logarithms into a single logarithm by division. For example, when we see \(\log_5 16 - \log 2\), we can rewrite it as \(\log_5 \frac{16}{2}\), simplifying our work.

Understanding these properties can save time and offer more straightforward solutions by reducing complex logarithmic expressions into simpler forms. This principle is not only applicable in exercises but is also useful in many real-world applications, such as calculating growth rates and signal processing.

Always remember these properties are valid only when the bases of the logarithms are the same, and the values are positive numbers, as logarithms of non-positive numbers are undefined.

Simplifying Expressions

Simplifying expressions is about making them easier to manage and understand. In the context of logarithms, simplifying expressions often involves combining terms using logarithm properties.

For instance, consider the expression: \(\log_5 16 - \log 2\). By applying the logarithmic property that allows us to subtract logs to get \(\log_5 \frac{16}{2}\), we simplify the expression to \(\log_5 8\).

Regularly practicing these simplifications strengthens understanding and enhances problem-solving efficiency. It adjusts focus from multiple cumbersome terms to a tidier, single expression. This streamlined approach can significantly reduce the likelihood of errors during calculations. Always simplify your expressions when possible to make manipulating and solving them more straightforward.

Equation Solving

Equation solving with logarithms involves using algebraic methods and logarithmic properties to find solutions. The ability to solve logarithmic equations effectively relies heavily on understanding both the properties of logarithms and skills in algebra.

When solving an equation like \(\log_5 16 - \log 2 = \log_5 8\), the crucial step is to utilize logarithmic properties to simplify the left-hand side. After simplification, if the equation reads \(\log_5 8 = \log_5 8\), it confirms the equation is true, as both sides are equal and have the same base.

Solving equations often involves these steps:

• Identify and apply appropriate logarithm properties.
• Simplify the expressions to direct comparisons.
• Check if both sides of the equation are equal after simplification.

Once you master these steps, solving logarithmic equations becomes a manageable task. Understanding these concepts simplifies the process as you encounter more complex logarithmic equations in mathematics.