Problem 40 - Precalculus

Question

Evaluate the logarithms exactly (if possible). $$\log 10^{-2}$$

Step-by-Step Solution

Answer

\( \log 10^{-2} = -2 \)

Step 1: Understand the Logarithm Base

The logarithm provided is \( \log 10^{-2} \). This is a logarithm with a base of 10, which is implied by the notation \( \log \) without any specified base. This simplifies calculations.

Step 2: Use the Logarithmic Identity

One of the basic properties of logarithms is \( \log_b(a^n) = n \cdot \log_b(a) \). Applying this property, we have \( \log_{10}(10^{-2}) = -2 \cdot \log_{10}(10) \).

Step 3: Calculate \( \log_{10}(10) \)

We know that \( \log_{10}(10) = 1 \) because any log base of its own number equals 1. This step simplifies our equation from Step 2.

Step 4: Complete the Calculation

Substitute the result from Step 3 into the expression from Step 2: \( \log_{10}(10^{-2}) = -2 \cdot 1 = -2 \).

Key Concepts

Understanding Logarithm Base

Logarithms are a way to express powers of numbers in a different form. In \(\log 10^{-2}\), the base of the logarithm is 10. When no base is specified, it's typically assumed to be 10, known as the common logarithm. The base of a logarithm is crucial because it determines the power to which the base must be raised to obtain a given number. This understanding allows us to perform operations like those in the original exercise with ease.

• The notation \(\log\) without an indicated base implies a base of 10.
• This makes the calculation much more straightforward when dealing with decimal-based numbers.
• Knowing the base helps in converting expressions into exponential form, simplifying the solving process.

This is why recognizing the base is the first crucial step in solving logarithmic problems, as demonstrated with the exercise.

The Magic of Logarithmic Identity

Logarithmic identities are powerful tools in simplifying complex logarithmic expressions. In the exercise, we utilized a vital identity: \(\log_b(a^n) = n \cdot \log_b(a)\). This identity states that when you have an exponent inside a logarithm, it can be brought out front as a multiplier. This principle simplifies calculations by breaking down exponents into a more manageable form.

• For example, \(\log_{10}(10^{-2})\) becomes \(-2 \cdot \log_{10}(10)\), making it easier to solve.
• This identity is helpful in scenarios where bases and exponents combine, and you need a straightforward way to dissect them.
• Such identities are essential in both simplifying calculations and in proving more complex relationships in mathematics.

Knowing and applying these identities can significantly reduce the effort required in solving logarithmic equations. They are a cornerstone in understanding the broader implications of logarithmic relationships.

Exploring Properties of Logarithms

Properties of logarithms, like the one used in solving \(\log 10^{-2}\), are fundamental tricks used to manipulate logarithmic expressions efficiently.
The property \((\log_{10}(10) = 1)\) highlights one of the most straightforward properties: a log base of its own immediate number always equals 1. This occurs because \(10^1 = 10\), and thus \(\log_{10}(10) = 1\). Applying this property simplified our original equation significantly, as seen in the step-by-step process.

• Understanding that \(\log_{b}(b) = 1\) is key in reducing lengthy calculations.
• These properties allow for quick verification and simplification of logarithmic expressions.
• They help transition from raw logarithmic expressions to more accessible numerical calculations.

By mastering these properties, one can quickly solve logarithmic problems without getting bogged down in complex computations, making these properties invaluable in both academic and practical applications.