Problem 36 - Intermediate Algebra

Question

For Problems \(21-40\), evaluate each expression. $$ \log _{5}\left(\frac{1}{25}\right) $$

Step-by-Step Solution

Answer

The value is -2.

Step 1: Understanding the Expression

The expression we need to evaluate is \( \log_{5}(\frac{1}{25}) \). We need to understand that this asks for the exponent to which 5 must be raised to yield \( \frac{1}{25} \).

Step 2: Expressing the Number in Terms of the Base

To solve the logarithm, express \( \frac{1}{25} \) as an exponent of 5. We know that \( 25 = 5^2 \). Therefore, \( \frac{1}{25} = \frac{1}{5^2} = 5^{-2} \).

Step 3: Applying the Logarithmic Identity

We can use the identity \( \log_{b}(b^x) = x \). Substituting into our expression, we have \( \log_{5}(5^{-2}) \). According to the identity, this simplifies to \( -2 \).

Step 4: Conclusion

Thus, the value of \( \log_{5}(\frac{1}{25}) \) is \(-2\).

Key Concepts

Exponents

Exponents are powerful tools in mathematics that represent repeated multiplication. When you see an expression like \( 5^2 \), it simply means 5 multiplied by itself once, which results in 25. The number 5 is called the base, and the number 2 is the exponent. Exponents can be negative as well. For instance, \( 5^{-2} \) means we take the reciprocal of \( 5^2 \), resulting in \( \frac{1}{25} \). When dealing with concepts like logarithms, understanding exponents is crucial as they lay the groundwork for grasping why we are trying to find what power a base is raised to achieve a given number. Exponents allow us to transform numbers to make calculations involving logarithms noticeably simpler.

Logarithmic Identities

Logarithmic identities are special rules that make working with logarithms easier. One key identity to know is \( \log_{b}(b^x) = x \). This states that if you have the logarithm of a number that's a power of the base (like \( b^x \)), the logarithm simplifies to just the exponent (\( x \)). For example, when you see \( \log_{5}(5^{-2}) \), you can quickly identify this as \( -2 \) because the base (5) is the same as in the exponent. Logarithmic identities like these help in swiftly reducing complex logarithmic expressions to simpler numbers or insights, providing clarity in mathematical problem solving.

Base Conversion

Base conversion is the process of changing the base of a logarithmic or exponential expression to make it more manageable. Sometimes, you encounter numbers or expressions that are not easily expressed in terms of the base you have. However, understanding base conversion helps you in evaluating logarithms without getting stuck. Take \( \log_{5}(\frac{1}{25}) \) as an example. Here, we focused on expressing \( \frac{1}{25} \) in terms of base 5 using exponents, which led us to \( 5^{-2} \). Once recognized, it simplifies moving forward with the problem. Observing how numbers can be expressed in different bases will give more flexibility and understanding when solving exponential or logarithmic problems. Base conversion is key to unlocking different doors in the mathematical world.