Problem 39 - Precalculus

Question

Evaluate the logarithms exactly (if possible). $$\log 10^{7}$$

Step-by-Step Solution

Answer

The exact value of \( \log 10^7 \) is 7.

Step 1: Identify the Logarithmic Base and Exponent

The expression we need to evaluate is \( \log 10^7 \). Recall that when no base is provided in a logarithmic expression, it is assumed to be base 10. This means we are working with \( \log_{10} 10^7 \).

Step 2: Apply the Properties of Logarithms

One important property of logarithms is that \( \log_b b^x = x \), where \( b \) is the base of the logarithm. In this case, \( b = 10 \) and \( x = 7 \). Apply this property to simplify: \( \log_{10} 10^7 = 7 \).

Key Concepts

Logarithmic Base

A logarithmic base is a fundamental part of understanding logarithms. In any logarithmic expression, like \( \log_b x \), the base \( b \) is the number that is raised to a power to produce \( x \). If your expression has no specified base, it’s commonly understood to be 10, known as the common logarithm. This is because base 10 logarithms are often used in calculations involving scientific notation and pH levels, among other applications.

Understanding your base is essential because it determines how you interpret a logarithm. For example, \( \log_2 8 = 3 \) means that 2 raised to the power of 3 gives 8. Therefore, identifying the correct base for \( \log 10^{7} \) ensures you know what values to work with in calculations. This exercise uses a base of 10, which makes the calculation straightforward thanks to the relationship between your base and exponents.

Exponents

Exponents tell us how many times to multiply a number by itself. In terms of logarithms, an exponent is the power to which the base must be raised to produce the "argument" of the log (the number you are taking the log of).

In our example with \( \log_{10} 10^7 \), the exponent is 7. This means that the base, which is 10, is raised to the power of 7. So, the exponent here shows that 10 multiplied by itself 7 times gives us the original number. This is a critical concept when translating between exponential and logarithmic forms since understanding exponents will guide you in simplifying logarithmic expressions. Recognizing that the base and the exponent are inherently linked simplifies the evaluation process.

Properties of Logarithms

The properties of logarithms are like rules that help simplify complex logarithmic expressions. One of the most crucial properties used in our example is:

• \( \log_b b^x = x \)

This means, when you take the logarithm of a base raised to an exponent, the result is just the exponent. This particular property works beautifully for \( \log_{10} 10^7 \) because it confirms that the solution is simply 7.

Logarithmic properties help convert multiplication, division, and exponents into simpler problems:

• Product Property: \( \log_b (xy) = \log_b x + \log_b y \)
• Quotient Property: \( \log_b \left(\frac{x}{y}\right) = \log_b x - \log_b y \)
• Power Property: \( \log_b x^y = y \cdot \log_b x \)

Familiarity with these rules not only speeds up calculations but also enhances understanding of logarithmic relationships. So by using the logarithmic property applied here, you can instantly recognize and solve problems such as \( \log 10^7 \).