Problem 21 - Algebra A Combined Function

Question

Use the power property to rewrite each expression. $$ \log _{5} \sqrt{y} $$

Step-by-Step Solution

Answer

\( \frac{1}{2} \cdot \log_5(y) \)

Step 1: Understand the Power Property of Logarithms

The power property of logarithms states that for any positive number \(a\) (where \(aeq1\)), and for all positive numbers \(M\), \(\log_a(M^n) = n \cdot \log_a(M)\). This property allows us to move exponents in an expression to the front of the logarithm as a multiplier.

Step 2: Express the Square Root as an Exponent

To apply the power property, we need to express the square root in terms of an exponent. Remember that the square root of \(y\) can be written as \( y^{1/2} \), which is the equivalent exponent form.

Step 3: Apply the Power Property

Now apply the power property of logarithms to the expression \( \log_5 (y^{1/2}) \). By the property, this becomes \( \frac{1}{2} \cdot \log_5(y) \), effectively moving the exponent \(1/2\) in front of the logarithm as a coefficient.

Key Concepts

Power Property of Logarithms

The power property of logarithms is a fundamental rule that helps simplify expressions involving logarithms and exponents. It is often used in algebraic manipulations to make computations easier. The property states that for any positive base, such as \( a \), and a positive number \( M \), the logarithm of \( M \) raised to a power \( n \) can be rewritten as a product:

• \( \log_a(M^n) = n \cdot \log_a(M) \)

This nifty property essentially enables you to "bring down" exponents outside the logarithm. It makes solving equations easier by converting an operation (raising to a power) inside the logarithm into a multiplication outside it.
For example, if you have \( \log_2(8^3) \), using the power property, this can be rewritten as \( 3 \cdot \log_2(8) \). It simplifies the process of evaluating or transforming logarithmic expressions.

Exponents

Exponents are a way to represent repeated multiplication of a number by itself. When a number \( x \) is raised to the power of \( n \), it simply means that \( x \) is multiplied by itself \( n \) times. This is written as \( x^n \).

• \( x^2 \) means \( x \times x \)
• \( x^3 \) means \( x \times x \times x \)

Exponents play a critical role in mathematics because they offer a shorthand notation for complex multiplications. They are also crucial when working with logarithms. In logarithmic contexts, exponents can often be transformed using rules like the power property, which simplifies more advanced operations. Understanding how exponents work, including fractional ones, can greatly assist in grasping logarithmic expressions.

Square Root as Exponent

The square root of a number can be expressed in exponential form. This involves using a fraction as an exponent. Specifically, the square root of any number \( y \) is the same as raising \( y \) to the power of \( \frac{1}{2} \).

• The expression \( \sqrt{y} \) can be written as \( y^{1/2} \).

Expressing square roots in exponential form is extremely useful when dealing with logarithms. This form allows you to apply logarithm rules, like the power property, more easily. Instead of working with roots directly, converting them into an exponent makes mathematical manipulations much simpler. This technique is widely used in calculus, complex number calculations, and many areas of algebra.

Logarithmic Expressions

Logarithmic expressions are mathematical statements that involve logarithms—functions that represent the inverse process of exponentiation. They are written as \( \log_b(a) \), which answers the question: "What exponent do we need to raise \( b \) to, in order to get \( a \)?"

• If \( b^x = a \), then \( \log_b(a) = x \).

Understanding logarithmic expressions is key to solving complex equations where the unknown variable appears as an exponent. They transform multiplicative processes into additive ones, making them particularly useful in various branches of mathematics, science, and engineering.
Logarithmic expressions can become more manageable by applying properties like the power property of logarithms, which transforms these expressions into simpler, equivalent forms, assisting in easier computations, derivations, and problem-solving activities.