Problem 503 - Precalculus

Question

Evaluate \(\log _{5}\left(\frac{1}{125}\right)\) without using a calculator.

Step-by-Step Solution

Answer

The value of \(\log_{5}\left(\frac{1}{125}\right)\) is -3.

Step 1: Understand the Definition

Recall that the logarithm \(\log_{b}(a)\) asks, "To what power must the base \(b\) be raised, to produce \(a\)?" In this problem, \(b = 5\) and \(a = \frac{1}{125}\).

Step 2: Express 125 as a Power of 5

Notice that 125 is a power of 5. Specifically, \(125 = 5^3\), meaning \(5^3 = 125\).

Step 3: Rewrite the Fraction

Since \(125 = 5^3\), the fraction \(\frac{1}{125}\) can be rewritten as \(5^{-3}\).

Step 4: Evaluate the Logarithm

Utilizing the property \(\log_{b}(b^x) = x\), substitute \(x = -3\) because \(5^{-3} = \frac{1}{125}\). Therefore, \(\log_{5}(\frac{1}{125}) = \log_{5}(5^{-3}) = -3\).

Key Concepts

Exponents

Exponents are a fundamental concept in mathematics that describe how many times we multiply a number by itself. For instance, when we say "5 raised to the power of 3," written as \(5^3\), we mean

• 5 is multiplied by itself 3 times.
• The result is \(5 \times 5 \times 5 = 125\).

Exponents are useful for
dealing with large numbers and simplifying mathematical expressions.
When you see a negative exponent, like \(5^{-3}\), it means:

• Take the reciprocal of the base raised to the positive exponent.
• In this case, \(5^{-3}\) means \(\frac{1}{5^3}\), which simplifies to \(\frac{1}{125}\).

Logarithmic Properties

Logarithms help us solve exponential equations by asking, "What exponent is needed to reach a certain number from a base?" They follow several key properties:

• For any base \(b\), \(\log_b(b) = 1\).
• The product rule: \(\log_b(mn) = \log_b(m) + \log_b(n)\).
• The power rule: \(\log_b(m^n) = n \cdot \log_b(m)\).
• The quotient rule: \(\log_b\left(\frac{m}{n}\right) = \log_b(m) - \log_b(n)\).

These properties are essential for simplifying complex logarithmic expressions.
For example, the power rule helps us break down expressions like \(\log_b(b^x) = x\) and find unknown exponents.
Understanding these rules unlocks the full potential of logarithms in problem-solving.

Base Recognition

The base in an exponential or logarithmic expression is the number that we repeatedly multiply. Recognizing bases when solving problems is crucial.
In the exercise, the base was 5. To evaluate \(\log_5(\frac{1}{125})\), we needed to express 125 as a power of the base, which happens to be 5 in this case.
The step-by-step breakdown involved realizing that \(125 = 5^3\). By identifying this, we could rewrite the fraction as \(\frac{1}{125} = 5^{-3}\).

• Base recognition simplifies logarithmic evaluations.
• It connects the dots between the base in a logarithm and the corresponding power expression.

When working with logarithms, always focus on identifying the base correctly.
This understanding will lead you to accurately express numbers in terms of their base powers.

Power Expressions

Power expressions describe numbers as repeated products. Recognizing power expressions is key to manipulating and understanding logs and exponents:

• To solve \(\log_5(\frac{1}{125})\), express 125 as \(5^3\).
• This becomes important because \(\frac{1}{125}\) is the same as \(5^{-3}\).
• Knowing this, applying the power drop property \(\log_b(b^x) = x\), simplifies our calculation to \(-3\).

Power expressions allow for concise mathematical communication.
They also simplify complex problems by expressing numbers with a common base, making calculations easier.
By converting to power expressions, you streamline solving logarithmic equations.