Question

Find the value of \(\left(7^{3}\right)^{-2 \log _{7} 8}:\) (a) \(8^{-7}\) (b) \(6^{-8}\) (c) \(8^{-6}\) (d) none of these

Short answer

Answer: (c) \(8^{-6}\)

Step-by-step solution

Apply the exponent rule (a^(m))^n = a^(m*n)

We will apply the exponent rule to the expression. For any numbers a, m, and n, (a^(m))^n = a^(m*n). Our expression becomes \((7^3)^{-2\log_7 8} = 7^{(3)(-2\log_7 8)}\).

Simplify the exponent

We will now simplify the exponent. \(3(-2\log_7 8) = -6\log_7 8\). Our expression becomes \(7^{-6\log_7 8}\).

Apply the logarithm rule a^(log_a b) = b

We apply the logarithm rule for any numbers a and b: a^(log_a b) = b. In our case, a=7 and b=8. Our expression simplifies to \(7^{(-6)\log_7 8} = (8)^{-6}\). Our simplified expression is \((8)^{-6}\). Comparing to the given answer choices, the correct choice is: (c) \(8^{-6}\)

Key concepts

Logarithmic Properties

Understanding logarithmic properties is crucial when dealing with various mathematical problems. These properties are like tools that help simplify complex logarithmic expressions, making them more manageable. Let's explore some of these properties that were used in our example exercise.

• Product Property: The logarithm of a product is the sum of the logarithms of its factors, expressed as
\( \log_a (bc) = \log_a b + \log_a c \).
• Quotient Property: The logarithm of a quotient is the difference of the logarithms, formulated as
\( \log_a (b/c) = \log_a b - \log_a c \).
• Power Property: The logarithm of a power is the exponent times the logarithm of the base, described by
\( \log_a (b^c) = c \cdot \log_a b \).
• Change of Base: The logarithm base change formula allows you to convert from one base to another, which is helpful when you have a base that is not convenient for calculations. It's given by
\( \log_b c = \frac{\log_a c}{\log_a b} \), where \( a \) is a new base.

These properties are especially useful when you are dealing with expressions involving logarithmic functions with a power, as in our example. The key is to recognize which property to use in order to simplify the expression efficiently.

Exponent Rules

Exponent rules or 'laws of exponents' are fundamental tools when working with expressions involving powers. These rules simplify the process of manipulating expressions and allow for clarity when dealing with complex equations. Here are some key exponent rules we've applied in solving our problem:

• Product of Powers: When multiplying two exponents with the same base, you can add the exponents, as in
\( a^m \cdot a^n = a^{m+n} \).
• Power of a Power: When taking an exponent to another power, you multiply the exponents, demonstrated by
\( (a^m)^n = a^{m \cdot n} \).
• Negative Exponent: A negative exponent indicates taking the reciprocal of the base and changing the sign of the exponent, like so
\( a^{-n} = \frac{1}{a^n} \).

In our example, we used the 'power of a power' rule to combine the exponents, which simplified our process greatly. It's important to be comfortable with these rules to recognize when they can effectively solve or simplify expressions.

Quantitative Aptitude

Quantitative aptitude covers the ability to handle numerical and mathematical calculations, and it's an essential skill for various competitive exams and everyday life problems. This skill is not just about knowing formulas but also about logical thinking and applying mathematical concepts effectively.

To excel in quantitative aptitude, one must be well-versed in areas such as arithmetic, algebra, geometry, and data interpretation. Regular practice is key to mastering these topics. Problem-solving techniques, such as simplifying complex equations, recognizing patterns, and understanding word problems, are also vital.

In solving our textbook example, quantitative aptitude comes into play by recognizing the relationship between the exponent rules and logarithmic properties in order to simplify expressions. This practical application of mathematical knowledge showcases a clear understanding of the topic and paves the way for solving a variety of problems with confidence and efficiency.