Problem 73 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use the properties of logarithms to condense the expression.$$4 \log _{3}(x+2)$$.

Step-by-Step Solution

Answer

The condensed form of the given logarithmic expression is \(\log_{3}((x+2)^4)\)

Step 1: Identify Coefficient

First, recognize the coefficient of the argument in the logarithmic function, which in this case is 4. It's attached to the logarithmic function \(\log_{3}(x+2)\).

Step 2: Apply Power Rule

Next, apply the power rule of logarithms. According to this rule, the coefficient 4 of the logarithm will become the exponent of the argument. So, you get: \(\log_{3}((x+2)^4)\).\nYou have now condensed the expression using logarithmic properties.

Key Concepts

Logarithmic Functions

Understanding logarithmic functions is crucial for solving various mathematical problems. These functions are the inverses of exponential functions and are represented as \( \log_b(x) \), where \(b\) is the base, and \(x\) is the argument of the log function. The base \(b\) must be a positive number, and not equal to 1, as the function would not be well-defined otherwise.

For example, when you come across \( \log_{3}(x+2) \), you're essentially looking for the power to which you need to raise 3 to get \(x+2\). If \( 3^y = x+2 \), then \( y = \log_{3}(x+2) \). This interchangeable relationship between exponential and logarithmic forms is the cornerstone of understanding logs and helps in various applications, such as solving exponential equations, analyzing data growth rates, or adjusting scales in scientific measurements.

Power Rule of Logarithms

The power rule of logarithms states that for any algebraic expression \(x\) and a positive real number \(n\), the expression \(n \log_{b}(x)\) can be rewritten as \(\log_{b}(x^n)\). Essentially, the power rule allows you to move the coefficient of a log function to the position of an exponent.

This property simplifies logarithmic expressions and is particularly useful when solving equations involving logs. Recall our initial example \(4 \log_{3}(x+2)\); by applying the power rule, it condenses down to \(\log_{3}((x+2)^4)\). The power rule is not only a theoretical concept but a practical tool that simplifies computation and is frequently used in fields such as biology, geology, and sound intensity measurement where logarithmic scales are commonplace.

Condensing Logarithmic Expressions

Condensing logarithmic expressions means to combine multiple log terms into a single term. This is achieved by using the properties of logarithms, such as the product, quotient, and power rules. It's very valuable when you want to make an expression simpler for solving or to prepare it for graphing or further manipulation.

When you're given the expression like the one in our original problem \(4 \log_{3}(x+2)\), condensing it makes it more manageable and closer to its core exponential form. After applying the appropriate rule, in this case, the power rule, it becomes \(\log_{3}((x+2)^4)\). The expression is no longer scattered across the coefficient and log but is neatly contained within a single logarithmic expression. It helps students and mathematicians alike to transition between logarithmic and exponential forms fluidly, which is an essential skill in understanding the behavior of logarithmic functions.