Question

Find the value of \(x\) for \(\log _{x} 2 . \log _{x / 16} 2=\log _{x / 64} 2:\) (a) 4 (b) 4,16 (c) 4,8 (d) \(4,8,32\)

Short answer

Answer: There is no valid solution for x within the given choices.

Step-by-step solution

Recalling the properties of logarithms

Before we start solving the equation, let's recall some logarithm properties that we will use: 1. \(\log_b(mn) = \log_b{m} + \log_b{n}\) 2. \(\log_b{\frac{m}{n}} = \log_b{m} - \log_b{n}\) 3. \(\log_b{m^n} = n\log_b{m}\)

Simplifying the given equation

We can simplify the given equation by utilizing property 3 from above: \(\log_{x}{2} \cdot (\log_{x}{2} - \log_{x}{16}) = \log_{\frac{x}{64}}{2} \cdot (log_{x}{2})\) Now, using property 1, we can get the following equation: \(\log_{x}{2} \cdot \log_{x}{2} - \log_{x}{2} \cdot \log_{x}{16} = \log_{\frac{x}{64}}{2} \cdot log_{x}{2}\)

Factoring out \(\log_{x}{2}\)

We can factor out \(\log_{x}{2}\) from both sides: \(\log_{x}{2} \cdot (\log_{x}{2} - \log_{x}{16}) = \log_{\frac{x}{64}}{2} \cdot log_{x}{2}\) Now, divide both sides by \(\log_{x}{2}\). Since \(\log_{x}{2}\) is not zero, this division is valid. Then, we will get: \(\log_{x}{2} - \log_{x}{16} = \log_{\frac{x}{64}}{2}\)

Simplifying further

Now, we apply property 2 to both sides and get: \(\log_{x}{\frac{2}{16}} = \log_{\frac{x}{64}}{2}\) Simplifying \(\frac{2}{16}\): \(\log_{x}{\frac{1}{8}} = \log_{\frac{x}{64}}{2}\)

Applying property 3 one more time

We'll apply property 3 again: \(-3\log_{x}{2} = \log_{\frac{x}{64}}{2}\)

Comparing the arguments

Since the bases of the logarithms are the same, we can compare the arguments: \(-3 = \frac{x}{64}\) Now, solving for \(x\): \(x = -192\) Since the base of any logarithm should be greater than 0 and not equal to 1, we discard this solution. Therefore, the given equation, \(\log_{x}{2} \cdot \log_{\frac{x}{16}}{2} = \log_{\frac{x}{64}}{2}\), has no solution in the given choices (a), (b), (c), and (d).

Key concepts

logarithmic equations

Logarithmic equations are a fundamental concept in algebra that involve logarithms. A logarithmic equation is an equation that involves the logarithm of a variable or number. In mathematical terms, it typically looks something like this: \( \log_b(x) \), where \( b \) is the base of the logarithm and \( x \) is the number we are taking the logarithm of. These equations are crucial in various fields, including mathematics, engineering, and the sciences.

To solve logarithmic equations, one often needs to use the properties of logarithms, including the product, quotient, and power rules. Understanding these properties aids in rewriting the equation in a form that makes it easier to solve. The goal is typically to use these properties to simplify and solve for the variable in the equation. A strong grasp of logarithmic principles can help solve more complex problems and handle real-world applications that involve exponential growth and decay.

When solving any logarithmic equation, it's essential to remember that the base of a logarithm must be greater than zero and not equal to one. Also, the argument of the logarithm (the value that \( x \) represents in \( \log_b(x) \)) must be positive. These conditions are crucial to ensure that the logarithm is defined.

properties of logarithms

The properties of logarithms are tools that make it easier to manipulate and solve logarithmic expressions. Here are some of the most important properties:

• Product Property: \( \log_b(mn) = \log_b{m} + \log_b{n} \) - This allows you to add the logarithms of two numbers when they are multiplied.
• Quotient Property: \( \log_b{\frac{m}{n}} = \log_b{m} - \log_b{n} \) - This allows you to subtract the logarithms of two numbers when they are divided.
• Power Property: \( \log_b{m^n} = n\log_b{m} \) - This allows you to multiply the logarithm by the exponent if the number is raised to a power.

These properties are useful for simplifying complex logarithmic expressions and equations, making it easier to isolate the variable. For example, in the given exercise, these properties were repeatedly used to simplify the equation into a manageable form.

Understanding and mastering these properties are key to solving logarithmic problems efficiently. It also helps in recognizing opportunities to simplify expressions or reveal hidden relationships between components of an equation.

solving logarithmic equations

Solving logarithmic equations often involves use of the discussed logarithmic properties to simplify and eventually isolate the variable. Here is a general approach to solving these equations:

Start by applying logarithmic properties to simplify the equation if it contains multiple logarithmic expressions. Try to combine or eliminate logarithms using the product, quotient, and power rules. If you have several terms involving logarithms within your equation, look to combine them into one.

• After simplification, aim to express the equation in terms of a single logarithm on one side. This way, you can convert the logarithmic equation into an exponential form. It is often easier to solve such equations by eliminating logarithms altogether.
• Finally, once in exponential form or when the equation is simplified, solve for the variable like any standard algebraic equation. Check for extraneous solutions that do not satisfy the requirements of a logarithm (i.e., arguments must be positive, and bases cannot be 1).

In cases where a direct solution is apparent, always reassess to confirm if these values satisfy the original logarithmic constraint. It is crucial to consider whether the compute is logical in the context of the problem. If not, there might be restrictions that suggest the equation has no practical solutions, as seen in the example exercise.

educational mathematics

Educational mathematics is about establishing a solid foundation of mathematical principles and practices. Understanding logarithms and logarithmic equations is an essential part of this education. They appear frequently in real-world applications like calculating compound interest, understanding the Richter scale in earthquakes, and more.

When learning mathematics, especially logarithms, the key is to practice regularly. Work through different problems to become comfortable with various properties and solution strategies. It's also beneficial to understand why certain steps are taken instead of just how they are done. Grasping the 'why' empowers you to tackle new problems with confidence.

• Math educators often recommend starting with simple logarithmic problems and progressively working towards more complex equations. This gradual advancement helps reinforce learning and ensures that the fundamental concepts are well understood.
• A variety of educational resources are available, such as textbooks, online courses, and interactive math platforms, to support learning outside of traditional classroom settings.

In summary, educational mathematics provides the tools and methods to approach logarithmic equations and other math concepts with ease. Understanding its principles and practicing regularly can lead to mastery of the subject, preparing students for academic tests and real-life scenarios alike.