Problem 58 - Algebra 2

Question

Write true or false for each statement. Justify your answer. \(\log _{2} 4+\log _{2} 8=5\)

Step-by-Step Solution

Answer

The statement is 'True'. The sum of logarithms \(\log _{2} 4 + \log _{2} 8\) indeed equals 5 because of the properties of logarithmic functions.

Step 1: Analyze and apply logarithm properties

According to the properties of logarithms, the sum \(\log _{2} 4 + \log _{2} 8\) can be rewritten as \( \log _{2} (4*8) \) because the base for both logs is the same.

Step 2: Simplify the equation

Calculate the product inside the log: \(4*8=32\). So, the logarithmic expression can be rewritten as \(\log _{2} 32\).

Step 3: Calculate the logarithm

Compute the value of the logarithmic expression. In this case, \(\log _{2} 32=5\) because 2 raised to the power 5 is equal to 32.

Step 4: Compare the obtained result with the given expression

Now compare the obtained result (5) with the numerical value stated in the initial problem, which is also 5. Since the numbers match, the equality is true.

Key Concepts

Properties of Logarithms

Logarithms have special properties that make them useful in various calculations. Understanding these properties can help simplify complex logarithmic expressions. One key property is the product rule for logarithms. It states that the logarithm of a product is equal to the sum of the logarithms of the individual factors. For example:

• If we have \( log_b (MN) = log_b M + log_b N \), then, for any valid base \(b\), the rule applies.

This property is particularly helpful when you need to combine multiple logarithms with the same base. It allows us to unite these into a single logarithm. Another important property is the power rule of logarithms. This states that the logarithm of a number raised to a power is equal to the power times the logarithm of the base number:

• For example, \(log_b (M^p) = p \cdot log_b M \)

There's also the quotient rule, which helps in dividing. It states:

• \( log_b \left(\frac{M}{N}\right) = log_b M - log_b N \)

These properties serve as tools to break down and manage logarithmic expressions more easily.

Logarithmic Expressions

Logarithmic expressions involve the use of logs to represent exponents in equations. Like other mathematical expressions, they can be simplified using the properties of logarithms. Let's take a closer look at the expression from the problem: \( log_2 4 + log_2 8 \).

• This represents the addition of two logarithms with the same base.

By applying the product rule, we can transform this into a single logarithm.
The sum is equivalent to the log of the product of the numbers inside the logs.
This means \( log_2 4 + log_2 8 = log_2 (4 \cdot 8) \), simplifying it to \( log_2 32 \).

• Such transformations can help solve equations or simply understand the problem better.

Transforming logarithmic expressions like this is crucial for solving many types of logarithmic equations and understanding their meaning in practical situations.

Simplifying Expressions

Simplifying expressions, particularly logarithmic ones, is a valuable skill in mathematics. It involves performing operations that make the expression easier to work with or more understandable.

• For instance, reducing \( log_2 4 + log_2 8 \) to \( log_2 32 \) through the product rule helps us quickly calculate:

To find the value of \( log_2 32 \), recognize that 32 can be expressed as a power of two.

• Since \(2^5 = 32\), it follows that \( log_2 32 = 5 \).

When simplifying, look for opportunities to transform the expression into a form that reveals the answer more directly.
This approach not only decreases the complexity but also increases the speed and efficiency of problem-solving. By practicing simplification in this way, you become better equipped to handle more challenging mathematical scenarios.