Question

Write an example that illustrates why $$ \log _{2}(x+y) \neq \log _{2} x+\log _{2} y $$

Short answer

For x = 2 and y = 4, \( \log_2 (2 + 4) eq \log_2 2 + \log_2 4 \).

Step-by-step solution

Choose specific values for x and y

Select simple values for x and y to make calculations easy. Let's use x = 2 and y = 4.

Calculate \( \log_2 (x + y) \)

Add x and y first: \( x + y = 2 + 4 = 6 \). Then find the base 2 logarithm of 6: \( \log_2 6 \).

Calculate \( \log_2 x + \log_2 y \)

Calculate the base 2 logarithm of x and y separately, then add them. \( \log_2 2 + \log_2 4 = 1 + 2 = 3 \).

Compare the results

Observe that \( \log_2 6 \) and \( \log_2 2 + \log_2 4 \) yield different results. Since \( \log_2 6 eq 3 \), it shows that \( \log_2 (x + y) eq \log_2 x + \log_2 y \).

Key concepts

logarithms

Logarithms are a powerful mathematical concept used to solve equations involving exponential growth or decay. A logarithm answers the question: 'To what power must we raise the base to obtain a specific number?'. For example, in the equation \textcolor{blue}{ \text{base}^\text{exponent} = \text{number} }, the logarithm would solve for the exponent. The notation is given as \textcolor{blue}{ \text{log}_\text{base}(number) = \text{exponent} }. Common bases are 10 (common logarithm) and \( e \) (natural logarithm), but logarithms can have any positive base.

base 2 logarithms

Base 2 logarithms are particularly important in computer science as they deal with binary systems. When we talk about \( \text{log}_2 \), we are asking: 'To what power must 2 be raised to result in a given number?'. For instance, \( \text{log}_2 8 = 3 \) because \textcolor{blue}{ 2^3 = 8 }. Understanding base 2 logarithms is crucial in algorithms, data structures, and network theories.

logarithmic identities

Logarithmic identities are rules that help simplify logarithmic expressions. Key identities include:

• \( \text{log}_a(\text{xy}) = \text{log}_a x + \text{log}_a y \)
• \( \text{log}_a(\frac{x}{y}) = \text{log}_a x - \text{log}_a y \)
• \( \text{log}_a(x^b) = b \text{log}_a x \)

In our exercise, we challenge the misconception that \( \text{log}_2(x + y) = \text{log}_2 x + \text{log}_2 y \). By using specific values for \( x \) and \( y \), we demonstrated that this is false. This reinforces the importance of correctly applying the sum and product properties of logarithms.

counterexamples in mathematics

Counterexamples are specific cases where a general statement is shown to be false. They are critical in disproving incorrect assumptions. In our problem, assigning \( x = 2 \) and \( y = 4 \), and then showing that \( \text{log}_2 (2 + 4) = \text{log}_2 6 eq 1 + 2 = 3 \), serves as a counterexample to the incorrect identity \( \text{log}_2(x + y) = \text{log}_2 x + \text{log}_2 y \). This highlights the need to be precise with logarithmic operations and understand their properties thoroughly.