Problem 69 - Algebra 2

Question

Write each expression as a single logarithm. \(7 \log _{10} p+\log _{10} q\)

Step-by-Step Solution

Answer

The expression simplified to a single logarithm is \( \log _{10} p^7q \)

Step 1: Apply the property of logarithms that allows us to multiply inside the log

With the logarithm property \(a \log_b n = \log_b n^a\), the expression can be rewritten as \( \log _{10} p^7+\log _{10} q \)

Step 2: Apply the property of logarithms that combines addition into multiplication

Using the second logarithm property \(\log_b m+\log_b n = \log_b(mn)\), the expression can be further simplified to \( \log _{10} p^7q \)

Key Concepts

Multiplication Rule

The multiplication rule in logarithms is an essential tool that allows you to simplify expressions with multiple logarithms. This rule helps us combine addition of two logarithmic expressions into one single expression. It is formulated as:

• For any logarithms with the same base, such as \(\log_b m + \log_b n = \log_b(mn)\), you can turn the sum into the log of a product.

For example, if you have \(\log_{10} 2 + \log_{10} 3\), according to the multiplication rule, it equals \(\log_{10}(2 \cdot 3)\). Therefore, it becomes \(\log_{10}6\).
This rule is very useful when you need to simplify expressions and work with them more effectively in math and science. Remember, that the multiplication rule only applies when the logarithms have the same base.

Exponentiation Rule

The exponentiation rule of logarithms is another powerful tool in simplifying complex logarithmic expressions. This rule helps to understand how multiplication by a constant outside a log correlates to an exponent inside the log. It's expressed as follows:

• For a given logarithm, \[a \log_b n = \log_b n^a\]\, you can bring the constant into the logarithm as an exponent of the argument.

Consider the expression \(7 \log_{10} p\). According to the exponentiation rule, you can rewrite this as \[\log_{10} p^7\].
This transformation simplifies calculations and is especially beneficial when dealing with large or complicated numbers. The exponentiation rule is commonly used in solving logarithmic equations and is an important concept to master for algebra and more advanced math subjects.

Logarithmic Expressions

Logarithmic expressions can sometimes seem complicated, but with the right properties, you can simplify them significantly. A logarithmic expression usually involves the logarithm of a number to a given base, such as \(\log_b n\).

• Logarithms essentially tell us what exponent we need to raise the base to get the number inside the logarithm.
• They are extremely useful in solving exponential equations and are a key concept in many fields, including science, engineering, and economics.

Consider the expression \(\log_{10} p + \log_{10} q\). Using the previously discussed properties, we can transform it into a single logarithmic expression: \(\log_{10} p^7q\).
Understanding these properties allows us to manipulate and solve equations to find unknown values easily. The goal with logarithmic expressions is to simplify them for easier computation and interpretation.