Question

Write each expression as a single logarithm. \(\log _{2}\left(\frac{1}{x}\right)+\log _{2}\left(\frac{1}{x^{2}}\right)\)

Short answer

\text{log}_2 \left( \frac{1}{x^3} \right).

Step-by-step solution

- Apply Logarithm Properties

Use the property of logarithms that states \(\text{log}_a (m) + \text{log}_a (n) = \text{log}_a (mn) \), which allows combining the two logarithms into one.

- Combine Logarithms

Combine \(\text{log}_2 \left( \frac{1}{x} \right) + \text{log}_2 \left( \frac{1}{x^2} \right)\) to get \(\text{log}_2 \left( \frac{1}{x} \cdot \frac{1}{x^2} \right)\).

- Simplify the Expression Inside the Logarithm

Simplify the expression: \(\text{log}_2 \left( \frac{1}{x \cdot x^2} \right) \ = \ \text{log}_2 \left( \frac{1}{x^3} \right).\)

Key concepts

Combining Logarithms

When you come across multiple logarithms in an expression, combining them can make the expression simpler. One of the useful properties is: \[\text{log}_a(m) + \text{log}_a(n) = \text{log}_a(mn)\] This helps merge two separate logarithms into one. For example, in our given exercise: \[\text{log}_2\bigg(\frac{1}{x}\bigg) + \text{log}_2\bigg(\frac{1}{x^2}\bigg)\] By applying the property, we combine them into: \[\text{log}_2\bigg(\frac{1}{x} \times \frac{1}{x^2}\bigg)\] This not only reduces the number of logarithmic terms but also sets up the expression for further simplification.

Logarithmic Simplification

After combining logarithms, the next step is simplification. We start with the combined expression: \[\text{log}_2\bigg(\frac{1}{x} \times \frac{1}{x^2}\bigg)\] Here, we multiply the fractions inside the logarithm. Multiplying fractions means multiplying their numerators and denominators: \[\frac{1}{x} \times \frac{1}{x^2} = \frac{1 \times 1}{x \times x^2} = \frac{1}{x^3}\] Therefore, the expression simplifies to: \[\text{log}_2\bigg(\frac{1}{x^3}\bigg)\] Simplifying logarithmic expressions makes them easier to understand and more usable in further algebraic manipulations.

Algebraic Manipulation

In algebra, manipulating expressions is crucial to solving problems. Once we have a simplified logarithmic expression like: \[\text{log}_2\bigg(\frac{1}{x^3}\bigg)\] We can further manipulate it using properties of exponents and logarithms. One such property is: \[\text{log}_a\bigg(\frac{1}{b}\bigg) = - \text{log}_a(b)\] Applying this to our expression, we get: \[\text{log}_2\bigg(\frac{1}{x^3}\bigg) = - \text{log}_2(x^3)\] Another useful property is: \[\text{log}_a(b^c) = c \times \text{log}_a(b)\] Using this, we can simplify further: \[- \text{log}_2(x^3) = -3 \times \text{log}_2(x)\] Through algebraic manipulation, we transform the original expression into a simple form that is easy to work with in additional equations or calculations.