Problem 52 - Precalculus : Building Concepts and Connections

Question

Use the definition of a logarithm to solve for \(x\). $$ \log _{5} \sqrt{5}=x$$

Step-by-Step Solution

Answer

\The value of \(x\) that satisfies the equation \( \log_{5}{\sqrt{5}} = x \) is 0.5.

Step 1: Convert the Square Root to an Exponent

We have \( \log _{5} \sqrt{5}=x\). A square root operation can be written as an exponent where the exponent is 0.5. So, the square root of 5 can be written as \(5^{0.5}\). That simplifies our equation to: \( \log_{5}{5^{0.5}} = x \).

Step 2: Use the Logarithm Rule

This rule states that logarithm base \(b\) of \(b^n\) is equal to \(n\), which can be written as \( \log_{b}{b^n} = n \). Applying this rule to our equation we get: \( 0.5 = x \).

Step 3: Present the Solution

Finally, we can present our result. The value of \(x\) that satisfies the equation \( \log_{5}{\sqrt{5}} = x \) is 0.5.

Key Concepts

Logarithm Properties

Logarithm properties are key techniques that simplify complex calculations. One such important property is the **logarithm power rule**. The power rule states that the logarithm of a base raised to an exponent can be simplified by multiplying the exponent with the logarithm of the base. Mathematically, it's expressed as

• \( \log_b{a^n} = n \cdot \log_b{a} \)

This property helps to break down exponential expressions into more manageable parts. In our original problem, the expression \( \log_{5}{5^{0.5}} \) uses this property.

Another useful property is the **identity property**, which is particularly straightforward.

• \( \log_b{b} = 1 \)

Because any number raised to the first power is itself. Therefore, when you have \( \log_{5}{5} \), it equals 1. These properties make solving logarithmic equations much easier.

Exponents

Exponents are a mathematical notation indicating the number of times a number, known as the base, is multiplied by itself. Thus, an exponent is fundamental in simplifying equations, particularly in logarithms!

Let's review some basic concepts:

• The base is the number being multiplied, and the exponent is the power to which the base is raised.
• If you have \(a^n\), \(a\) is the base, and \(n\) is the exponent.
• A square root, like \(\sqrt{5}\), can be expressed as an exponent: \(5^{0.5}\).

This conversion is very useful. Transforming roots into fractional exponents makes it easier to apply logarithmic properties.

Understanding exponents aids in interpreting and solving many logarithmic equations, enabling the use of logarithm properties effectively.

Mathematical Equations

Mathematical equations are statements asserting that two expressions are equal, using the equality symbol \(=\). Solving an equation involves manipulating it to find the value of unknowns.

Consider the logarithmic equation:

• \( \log _{5} \sqrt{5} = x \)

To solve such equations, clear understanding of both exponents and logarithms is crucial. By expressing \(\sqrt{5}\) as \(5^{0.5}\), the equation becomes manageable:

• \( \log_{5}{5^{0.5}} = x \)

Using the property \( \log_b{b^n} = n \), it simplifies to \(x = 0.5\).

Such transformation highlights how solving equations involves a step-by-step approach. Identifying modifications that simplify expressions is vital for finding solutions efficiently.