Problem 48 - College Algebra with Corequisite Support

Question

For the following exercises, evaluate the common logarithmic expression without using a calculator. \(\log (1)+7\)

Step-by-Step Solution

Answer

The value of \( \log(1) + 7 \) is 7.

Step 1: Evaluate log(1)

The common logarithm of 1, denoted as \( \log(1) \), is the power to which the base 10 must be raised to result in 1. Since any non-zero number raised to the power of 0 is 1, it follows that \( \log(1) = 0 \).

Step 2: Substitute log(1) with 0

Replace \( \log(1) \) with 0 in the original expression: \( 0 + 7 \).

Step 3: Simplify the Expression

Add 0 to 7, as 0 is the additive identity, which means adding 0 to any number does not change its value. Thus, \( 0 + 7 = 7 \).

Key Concepts

Logarithmic Expressions

Logarithmic expressions involve the use of logarithms within mathematical statements or equations. A logarithm essentially asks the question, "To what exponent must we raise the base to get a certain number?" For example, in the expression \( \log_{10}(1) \), we are determining what power you raise the base 10 to produce the number 1.

Logarithmic expressions can appear in various forms and can involve multiple terms.

• Often, these expressions combine different mathematical operations, like addition or subtraction, similar to the one in our problem: \( \log (1) + 7 \).
• Such expressions require evaluating each logarithmic term first before proceeding with any arithmetic operations.

Evaluating these expressions carefully requires an understanding of the specific properties and rules of logarithms, which guide how these expressions are expanded, simplified, or solved.

Base 10 Logarithm

The base 10 logarithm, commonly referred to as the "common logarithm," uses 10 as its base. This base is particularly useful in scientific calculations and is standard in many fields like engineering and finance. When using a base 10 logarithm, the expression is typically written simply as \( \log(x) \) instead of \( \log_{10}(x) \).

A key feature of base 10 logarithms is their application to numbers that are powers of 10.

• For instance, \( \log(10^3) = 3 \) because 10 must be raised to the power of 3 to produce 1000.
• This makes the common logarithm quite intuitive for estimating the size of numbers in terms of their decimal composition.

Understanding base 10 helps with manually evaluating expressions such as \( \log(1) \), where it is known that because any number raised to the power of 0 equals 1, the result is always 0. In more complex cases, the base 10 logarithm allows for the simplification and assessment of numbers without resorting to a calculator.

Properties of Logarithms

Properties of logarithms are a set of rules that facilitate the manipulation and simplification of logarithmic expressions. These properties are derived from the characteristics of exponents, as logarithms are inherently linked to exponential functions. Some common properties include:

• Product Property: \( \log_b(MN) = \log_b(M) + \log_b(N) \) which decomposes the log of a product into the sum of two logs.
• Quotient Property: \( \log_b \left(\frac{M}{N}\right) = \log_b(M) - \log_b(N) \) which breaks down the log of a division into a subtraction.
• Power Property: \( \log_b(M^p) = p \cdot \log_b(M) \) where the exponent can be 'pulled out' as a multiplier.

In the context of our original problem, knowing the property that \( \log(1) = 0 \) is crucial. This is due to the power property of logarithms, since any number raised to the zero power equals one.

By applying these properties effectively, it becomes possible to evaluate and simplify logarithmic expressions more efficiently, either by hand or when using additional mathematical operations.