Problem 12 - Precalculus

Question

Apply the properties of logarithms to simplify each expression. Do not use a calculator. $$2^{\log _{2} 5}$$

Step-by-Step Solution

Answer

The simplified expression is 5.

Step 1: Understanding the Problem

We are given the expression \(2^{\log_{2} 5}\) and asked to simplify it using properties of logarithms. This involves recognizing how logarithms and exponents can be combined.

Step 2: Apply the Power Rule of Logarithms

The power rule of logarithms is \(a^{\log_{a}(x)} = x\). Since the base of the exponent and the base of the logarithm are the same in our expression, we can directly apply this rule.

Step 3: Implement the Power Rule

By applying the power rule \(a^{\log_{a}(x)} = x\) to our expression, we identify \(a = 2\) and \(x = 5\). Therefore, \(2^{\log_{2}5} = 5\).

Step 4: Conclusion

The expression \(2^{\log_{2} 5}\) simplifies directly to 5 using the property of logarithms. This is a straightforward application of the power rule with matching bases.

Key Concepts

Exponents

An exponent refers to the number of times a number, known as the base, is multiplied by itself. For instance, in the expression \(a^n\), \(a\) is the base, and \(n\) is the exponent. Understanding exponents is crucial because they simplify expressions involving repeated multiplication.

• The number \(a^2\) means \(a\) multiplied by itself once: \(a \times a\).
• Similarly, \(a^3\) refers to \(a \times a \times a\).

In a broader sense, exponents help manage large numbers, especially in scientific calculations where they are often used to denote powers of ten. By mastering exponents, you open the door to understanding complex mathematical concepts and making calculations more manageable.

Power Rule

The power rule is a property of logarithms that provides an essential shortcut for simplifying expressions where the base and logarithm are the same. The rule states that for any positive number \(a\) and any real number \(x\), then \(a^{\log_{a}(x)} = x\).
This rule works because it combines the operations of exponents and logarithms in a way that effectively "cancels out" their effects.

• The expression \(a^{\log_{a}(x)}\) read as "\(a\) raised to the power log base \(a\) of \(x\)," results in \(x\).
• This happens because logarithms answer the question, "To what power must the base \(a\) be raised, to produce \(x\)?"

By recognizing this rule, problems that seem complex at first can be reduced to simpler forms. This principle is wonderfully demonstrated in the expression \(2^{\log_{2}5}\), which directly simplifies to \(5\) using the power rule.

Simplifying Expressions

Simplifying expressions involves reducing a mathematical statement to its simplest form. This process can make calculations easier and solutions more intuitive.
Key steps include:

• Identifying and applying relevant properties of operations, such as the power rule.
• Canceling terms that are opposites or inverses, like simplifying \(a^{\log_{a}(x)}\) to \(x\).
• Looking for patterns or standardized forms to use established shortcuts.

For the expression \(2^{\log_{2} 5}\), simplification involves recognizing the pattern that matches the power rule.
Using such techniques not only helps in solving algebraic equations efficiently but also builds a sound foundation for advanced mathematics by focusing on understanding and applying core mathematical properties.