Problem 26 - Algebra A Combined Function

Question

Find the exact value of each logarithm. $$ \log 10^{7} $$

Step-by-Step Solution

Answer

The exact value is 7.

Step 1: Identify Properties

First, recognize the logarithm property that states \[ \log_b (b^x) = x \]for any base \(b\) and exponent \(x\). In this case, the base is 10 and the exponent is 7.

Step 2: Apply the Logarithm Property

Using the property mentioned, apply it to the given logarithm:\[ \log_{10} (10^7) = 7 \]Since the base \(b\) in the logarithm is the same as the base of the exponent, the logarithm simplifies to just the exponent.

Step 3: Confirmation

Check by rewriting the problem using the property.Since the expression \(\log 10^7\) is simplified by recognizing that the function and the exponent cancel out, we confirm:\[ \log_{10}(10^7) = 7 \]

Key Concepts

Logarithm Properties

The concept of logarithm properties is fundamental in simplifying complex logarithmic expressions. One of the key properties is the power rule, which states:

• \( \log_b (b^x) = x \) for any given base \(b\)
• This property tells us that when a logarithm has the same base as the power inside the log, the result is simply the exponent \(x\)

Understanding this property can make calculations more manageable by eliminating potential confusion and unnecessary steps in solving logarithmic equations.
For example, in the problem \( \log_{10} (10^7) \), we immediately see that the base of the logarithm matches the base of the exponent, simplifying directly to 7. Recognizing these properties helps in quickly and efficiently finding solutions.

Base of Logarithm

The base of a logarithm is a crucial component that specifies the number system from which the logarithm is derived. In mathematical expressions, the base is written as a small number just below the \'log\' symbol:
\( \log_b A \), where \(b\) is the base and \(A\) is the argument of the logarithm.

Here are some key points about the base of logarithms:

• Commonly used bases include 10 (common logarithm), \(e\) (natural logarithm), and 2 (binary logarithm)
• The choice of base affects the output; thus, changing the base can modify the computation

Ensure consistency in using bases when solving problems to avoid errors. In some cases, the base may not be displayed explicitly; for example, when base 10 is used, it can be simply shown as \( \log \) with no number. This is understood to mean the common logarithm with base 10.

Exponents and Logarithms

Exponents and logarithms are inverse operations, much like addition and subtraction, or multiplication and division. This means they can undo each other, offering a pathway to solve equations involving powers:

• When \( b^x = A \), the logarithmic form is \( x = \log_b A \)
• This relationship can be used interchangeably between exponential and logarithmic forms; solving for one \'undoes\' the other

This inverse nature is what allows properties such as \( \log_b (b^x) = x \) to be applicable and useful. By applying these principles, the problem of finding the value of a logarithmic expression can be simplified considerably. When given \( \log_{10} (10^7) \), it follows from the inverse nature that the expression simplifies directly to the exponent, 7.
This connection between exponents and logarithms is essential for more advanced mathematics topics, and learning to switch between these forms can provide deeper insight into algebraic principles.