Question

If \(u=v^{2}=w^{3}=z^{4}\), then \(\log _{u}(u v w z)\) is equal to : (a) \(1+\frac{1}{2}-\frac{1}{3}-\frac{1}{4}\) (b) 24 (c) \(1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}\) (d) \(\frac{1}{24}\)

Short answer

Answer: The value of \(\log_u(uvwz)\) is \(1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}\).

Step-by-step solution

Rewrite \(v\), \(w\), and \(z\) in terms of \(u\)

Since we are given \(u=v^2=w^3=z^4\), we can rewrite the expressions for \(v, w\) and \(z\) in terms of \(u\): \[v = \sqrt{u}\] \[w = \sqrt[3]{u}\] \[z = \sqrt[4]{u}\]

Substitute the values into the expression

Now, we'll substitute these values back into the expression we need to find, \(\log_u(uvwz)\): \[\log_u\left(u\cdot\sqrt{u}\cdot\sqrt[3]{u}\cdot\sqrt[4]{u}\right)\]

Simplify the expression

Next, we want to simplify the expression inside the logarithm: \[u\cdot\sqrt{u}\cdot\sqrt[3]{u}\cdot\sqrt[4]{u} = u \cdot u^{\frac{1}{2}} \cdot u^{\frac{1}{3}} \cdot u^{\frac{1}{4}}\] Since we are multiplying the same base with different exponents, we can add the exponents as a property of exponents: \[u \cdot u^{\frac{1}{2}} \cdot u^{\frac{1}{3}} \cdot u^{\frac{1}{4}} = u^{1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4}}\]

Evaluate the logarithm

Now we can substitute this back into the log expression and evaluate: \[\log_u\left(u^{1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4}}\right)\] The log base \(u\) of any number \(u^x\) is equal to the exponent \(x\). Therefore, \[\log_u\left(u^{1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4}}\right) = 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4}\]

Match the answer to the given options

Matching our answer from step 4 to the given options, we find that our answer matches with option (c). So, the correct answer is: \[\log_u(uvwz) = 1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}\]

Key concepts

Exponent Properties

Understanding the behavior of exponents is fundamental in mathematics, especially when dealing with more complex equations involving variables and their powers. In the problem provided, the concept of exponent properties plays a critical role.

Exponents, also known as powers, indicate how many times a number, the base, is multiplied by itself. A few rules govern the operations involving exponents:

• The product of powers rule states that when multiplying two powers with the same base, you add their exponents, as in \(a^m \times a^n = a^{m+n}\).
• The power of a power rule dictates that when you have a power raised to another power, you multiply the exponents, resulting in \( (a^m)^n = a^{mn} \).
• For the power of a product rule, when a product is raised to a power, the exponent is distributed to each factor within the parentheses, as in \( (ab)^n = a^n b^n \).

These properties are intricately used when simplifying logarithmic expressions, allowing us to combine terms and reduce expressions to a more manageable form, as shown in the solution by summing the exponents of the variable \(u\).

Logarithmic Expressions

Logarithmic expressions are mathematical statements that involve logarithms, which are the inverses of exponential functions. A logarithm, written as \(\log_b(a)\), asks the question: 'To what power must I raise \(b\) to get \(a\)?' The answer to this is the actual value of the log.

In the problem at hand, we see the term \(\log_u(uvwz)\). This is a logarithmic expression where \(u\) is the base and \(uvwz\) is the argument. The power to which the base \(u\) must be raised to produce the argument is what the logarithm equals. When the argument of the log is an expression involving multiplication of different powers of the same base, the property that \(\log_b(mn) = \log_b(m) + \log_b(n)\) can be used, which is a consequence of the fundamental logarithmic identity that relates multiplication inside the log to addition of logs. This explains why simplified expressions within a log are critical for straightforward calculations.

Simplifying Logarithms

Simplifying logarithms is a key skill in algebra to make complex logarithmic expressions more manageable. This involves using exponent properties within the argument of the logarithm and applying basic log rules.

For example, when faced with a logarithm of a product as in our exercise, we can use exponent properties to combine the terms and simplify the expression inside the log before we apply the logarithm. This simplification process aligns the expression with one of the basic log rules that the \(\log\) of an exponentiated base (\(\log_b(b^x)\)) is simply the exponent \(x\). The ability to simplify the argument to the form \(u^x\) for some exponent \(x\) is crucial because it often leads directly to the final answer when evaluating the logarithm, as seen when the given expression transformed to \(u^{1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4}}\) before the log operation was applied.

Quantitative Aptitude

Quantitative aptitude encompasses the ability to handle numerical and logical reasoning effectively, an essential skill in a wide range of academic fields, competitive exams, and general problem-solving. It involves understanding numerical concepts, operations and their applications.

In the context of logarithm problems, quantitative aptitude refers to the ability to approach and solve numerical problems involving logarithms with precision and in an optimized manner. It involves recognizing the patterns, such as exponent properties, and applying them appropriately to simplify logarithmic expressions. Increasing one's quantitative aptitude in mathematics includes familiarizing oneself with a variety of problem types and practicing the specific techniques required to address them efficiently, like those showcased in solving the exercise. Problems that appear complex at first can often be broken down into simpler components through a step-by-step approach, applying fundamental math concepts to reach a solution.