Problem 17 - Precalculus with Limits

Question

Using Properties of Logarithms In Exercises \(15-20\) , use the properties of logarithms to rewrite and simplify the logarithmic expression. $$\log _{5} \frac{1}{250}$$

Step-by-Step Solution

Answer

After simplifying the expression \(\log_{5}\frac{1}{250}\), we get that the final answer is -4.

Step 1: Identify the Expression

First, identify the given expression \( \log _{5} \frac{1}{250}\). This is a logarithm with base 5 and argument \(\frac{1}{250}\)

Step 2: Apply the Property of Logarithm

Next, the property of logarithm that allows us to simplify this term is \(\log_{b}{\frac{1}{a}} = -\log_{b}{a}\). Applying this property, this simplifies our expression to \( -\log_{5}{250}\)

Step 3: Simplify the Expression

We now have \( -\log _{5} 250\), this is equivalent to finding the exponent to which we have to raise 5 to get 250. Since \(5^3 = 125\) and doubling 125 gives 250, and as a result, the exponent is now 4. So it simplifies to \( -4 \)

Key Concepts

Properties of Logarithms

Logarithms have several properties that help simplify expressions and make calculations easier. These properties include the Product Property, the Quotient Property, and the Power Property. In this exercise, the focus is on specific properties useful when dealing with division and negative exponents.

• The **Quotient Property** states that the logarithm of a quotient is the difference of the logarithms: \( \log_b \frac{x}{y} = \log_b x - \log_b y \).
• An additional useful property is the **Negative Exponent Property**, which translates the reciprocal in a logarithm into a negative sign: \( \log_b \frac{1}{x} = -\log_b x \). This is crucial for dealing with expressions like \( \log_{5} \frac{1}{250} \) which was simplified using this property.

Understanding these properties allows you to break down complex expressions into more manageable pieces, making it easier to calculate or approximate their values. Applying these in different scenarios can vastly reduce the effort needed in solving exercises.

Simplifying Logarithmic Expressions

When simplifying logarithmic expressions, we often aim to express them in the simplest form possible using known rules or equivalent expressions. Let's consider our given expression: \( \log_{5} \frac{1}{250} \).Firstly, identify any properties of logarithms that can be applied. For this expression, we apply the **Negative Exponent Property** as mentioned above, simplifying it to \( -\log_{5} 250 \).The next step involves solving \( \log_{5} 250 \). Since there's no handy power of 5 that exactly equals 250, some mental math can help. Recognize that \( 5^3 = 125 \), and double that to approximate \( 5^4 \) reaching 625, which is much higher. Calculate between these known powers to approximate more accurately. Through this process, you determine the expression to equal \( -4 \), showing the initial logarithm translates to \(-4\) in simplest form.This approach makes use of identifying appropriate properties and solving step by step, ensuring all expansions, substitutions, or transformations are correct and clear.

Base Conversion in Logarithms

Base conversion in logarithms is all about changing the base of a given logarithm while keeping the expression equivalent. This is particularly useful when you have to solve logarithms with uncommon bases or when using a different base simplifies calculations.The main formula for base conversion is: \[ \log_b x = \frac{\log_k x}{\log_k b}\]Where \(k\) can be any base you choose, often 10 or \(e\) for easier calculation using calculators because these are "common" and "natural" logs, respectively.Suppose you want to convert \( \log_{5} 250 \) into base 10 logs, it could become \( \frac{\log_{10} 250}{\log_{10} 5} \). This change helps when logging tools handle common bases more effectively, or if you need to align bases with other parts of an expression.Understanding this property opens up ways to solve complex problems by switching to more familiar bases and can sometimes simplify computation significantly when working with different logarithmic functions.