Problem 45 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use the properties of logarithms to verify the equation.$$\log _{5} \frac{1}{250}=-3-\log _{5} 2$$.

Step-by-Step Solution

Answer

The given equation \(\log _{5} \frac{1}{250}=-3-\log _{5} 2\) is evaluated as true by step-by-step manipulation and application of logarithm properties.

Step 1: Apply the power rule of logarithm

Let's first use the power rule of logarithm which states, if \(\log _{a} x=n\), then \(x=a^n\). Therefore, we rewrite \(\log _{5} \frac{1}{250}\) as \(5^{-3}\). Now the equation becomes: \(5^{-3} = -3-\log _{5} 2\).

Step 2: Transform the equation

Next, we transform this equation to isolate the logarithmic term on one side of the equation. So, we add 3 to both sides of the equation which yields: \(5^{-3}+3 = -\log _{5} 2\). This simplifies to: \(3-\frac{1}{125} = -\log _{5} 2\).

Step 3: Apply the third logarithm property

Now we apply the third logarithm property which states that \(\log _{a} \frac{b}{c}= \log _{a} b - \log _{a} c\). Thus, we convert the left side of the equation back to a logarithmic form: \(\log _{5} \frac{3125}{125} = -\log _{5} 2\). This simplifies to: \(\log _{5} 25 = -\log _{5} 2\).

Step 4: Rearrange the equation

Finally, we arrange this equation to be similar to the given equation. To do that, we multiply both sides of the equation by -1. Hence, \(-\log _{5} 25 = \log _{5} 2\). So, if we rewrite -25 as \(\frac{1}{5^2}\) or \(\frac{1}{25}\), the equation becomes \(\log _{5} \frac{1}{25} = \log _{5} 2\), which is true, thus verifying the given equation.

Key Concepts

Power Rule of Logarithms

The power rule of logarithms is a useful tool when dealing with logarithmic expressions. This rule states that if you have a logarithm of a number raised to an exponent, you can simplify it by multiplying the exponent by the logarithm of the base. Mathematically, it is expressed as \(\log_{a}(x^n) = n \cdot \log_{a}(x)\). This rule helps in breaking down complex expressions, and in this particular exercise, it aids in transforming expressions like \(\log_{5} \frac{1}{250}\) to more manageable forms by identifying that \(\frac{1}{250} = 5^{-3}\).

Understanding the power rule can greatly simplify solving logarithmic equations and converting them into simpler forms. It allows you to bring down large exponents, making the equations easier to manage. In step 1 of the exercise, we applied this by rewriting \(\log_{5} \frac{1}{250}\) as \(5^{-3}\).

Remember, this tool makes it easier to handle equations where numbers are expressed as powers of a base, giving more control over how to manipulate and solve equations effectively.

Properties of Logarithms

Logarithms possess several useful properties that allow us to manipulate and solve logarithmic expressions effectively. These properties are essential for transforming and simplifying equations.

One important property is the product rule: \(\log_{a}(bc) = \log_{a}(b) + \log_{a}(c)\). This helps in expanding logarithms.

Conversely, the quotient rule states: \(\log_{a}\left(\frac{b}{c}\right) = \log_{a}(b) - \log_{a}(c)\). In the exercise, we used this property to convert expressions on different sides of the equation, showcasing how subtracting logs can be turned into a single logarithm of division.

• The power rule: already discussed, is another invaluable asset when breaking down exponents.

Applying these properties allows us to rearrange equations, helping in isolating the logarithmic term and transforming the equation as executed in step 3 of this exercise.

By understanding these properties, students improve their ability to handle logarithmic expressions in various mathematical contexts.

Logarithmic Equations

Logarithmic equations are equations that involve the logarithm of a variable or a number. Solving these equations often involves using properties of logarithms, as well as converting between logarithmic and exponential forms.

In general, solving a logarithmic equation involves:

• Using the properties of logarithms to simplify or transform the equation;
• Exponentiating the log form to solve for the variable;
• Checking the solutions to ensure they make sense within the given equation.

To solve the equation given in the exercise, it's crucial to carefully apply properties such as the power rule and the quotient rule, which help in simplifying and rearranging the terms. Step-by-step transformation is key, as shown in steps 1-4 of the solution.

Mastering the art of solving logarithmic equations is a valuable skill, as it extends to various fields such as finance, engineering, and science where exponential growth or decay is analyzed and understood. This understanding aids in accurately predicting and solving problems encountered in these fields.