Problem 72 - Intermediate Algebra

Question

Find each value of \(x .\) $$ \log _{25} x=\frac{1}{2} $$

Step-by-Step Solution

Answer

The value of \( x \) is 5.

Step 1: Understand Logarithm Definition

The equation given is \( \log_{25} x = \frac{1}{2} \). This means finding the value of \( x \) such that raising 25 to the power of \( \frac{1}{2} \) gives \( x \). Hence, the equivalent exponential form is \( 25^{\frac{1}{2}} = x \).

Step 2: Simplify the Exponent

The expression \( 25^{\frac{1}{2}} \) represents the square root of 25. Therefore, we need to evaluate \( \sqrt{25} \).

Step 3: Calculate the Square Root

The square root of 25 is 5. Therefore \( \sqrt{25} = 5 \).

Step 4: Finalize the Solution

Substituting back, \( x = 5 \). Hence, the value of \( x \) is 5.

Key Concepts

Exponential Functions

Exponential functions are mathematical expressions where a constant base is raised to a variable exponent. These functions are integral to many areas in mathematics and science, as they describe growth and decay processes such as population growth or radioactive decay. In the exercise, we see the concept of reversing a logarithm into its exponential form.The equation \( \log_{25} x = \frac{1}{2} \) translates into the exponential equation \( 25^{\frac{1}{2}} = x \). Here, the base is 25, and the exponent is either a whole number or a fraction, in this case, \( \frac{1}{2} \). Such expressions with fractions often point to the concept of roots. Adjusting base and exponents helps find the exact values for unknowns in algebraic problems, making exponential functions a crucial tool in mathematical analysis.- **Key Points:** - Exponential functions convert logarithmic expressions. - The base raised to the exponent gives us the desired value. - These functions often involve fractional exponents indicating roots.

Square Roots

Square roots are operations that find a value which, when multiplied by itself, gives the original number. They are fundamental in solving equations involving exponential functions, especially when dealing with fractional exponents.In the given exercise, when interpreting \( 25^{\frac{1}{2}} \), it's essential to realize this means we seek a number that, when squared, equals 25. Therefore, \( \sqrt{25} = 5 \) because 5 multiplied by 5 equals 25.Square roots provide valuable simplification in algebraic expressions and help in understanding the nature of numbers and equations. Considering roots helps break down complex equations into easily manageable parts, ensuring a more straightforward path to the correct solution.- **Key Points:** - Square root of a number \( x \) is the number that when squared equals \( x \). - Fractional exponents translated into roots simplify equations. - \( \sqrt{25} = 5 \) because \( 5 \times 5 = 25 \). Understanding this operation is crucial for solutions involving roots.

Equivalent Forms

Equivalent forms in mathematics refer to different ways of expressing the same value or relationship. It's a common technique used to simplify complex equations and to make equations more understandable or easier to work with.In this context, we started with a logarithmic equation \( \log_{25} x = \frac{1}{2} \) and converted it into an exponential form \( 25^{\frac{1}{2}} = x \). This conversion shows that different mathematical operations or functions can represent the same relationship.By utilizing equivalent forms, we can switch between forms to leverage specific mathematical strategies or properties that make solving equations more straightforward. Understanding equivalent forms is crucial since it allows flexibility and enhances problem-solving skills by seeing the connection between different mathematical concepts.- **Key Points:** - Multiple forms can represent the same mathematical relationship. - Conversions between forms, such as logarithms to exponents, clarify solutions. - Recognizing and using equivalent forms enhances understanding and problem-solving.