Question

If \(a^{3-x} b^{5 x}=a^{x+5} b^{3 x}\), then the value of \(x \log \left(\frac{b}{a}\right)\) is: (a) \(\log a\) (b) \(\log b\) (c) \(\log x\) (d) 1

Short answer

Answer: (a) \(\log a\)

Step-by-step solution

Simplify the given equation using properties of exponents

Given: \(a^{3-x} b^{5 x}=a^{x+5} b^{3 x}\). We will use properties of exponents to simplify the equation. Divide both sides by \(a^{3-x} b^{3 x}\): \(\frac{a^{3-x} b^{5 x}}{a^{3-x} b^{3 x}}= \frac{a^{x+5} b^{3 x}}{a^{3-x} b^{3 x}}\) After simplification, we get: \(b^{2x} = a^{2x+2}\)

Take the logarithm of both sides

To further simplify, take the logarithm of both sides: \(\log (b^{2x}) = \log (a^{2x+2})\)

Use properties of logarithms to simplify

Using properties of logarithms: \(2x (\log b) = (2x+2) (\log a)\) Now, let's isolate \(x\) on one side: \(2x \log b - 2x \log a = 2 \log a\)

Factor out the \(2x\)

Factor out a \(2x\) from the left-hand side: \(2x(\log b - \log a) = 2 \log a\)

Solve for \(x \log\left(\frac{b}{a}\right)\)

As per the question, we need to find the value of \(x \log\left(\frac{b}{a}\right)\). Using the property of logarithms, \(\log b - \log a = \log\left(\frac{b}{a}\right)\), we have: \(2x \log\left(\frac{b}{a}\right) = 2 \log a\) Now, we are required to find the value of \(x \log\left(\frac{b}{a}\right)\), which we can now obtain by dividing both sides by 2: \(x \log\left(\frac{b}{a}\right) = \log a\) The value of \(x \log\left(\frac{b}{a}\right)\) is \(\log a\). Therefore, the correct answer is (a) \(\log a\).

Key concepts

Properties of Exponents

Exponents are a way to represent repeated multiplication. The properties of exponents make it simpler to manipulate and solve equations involving powers. Here are some useful properties:

• Product of Powers: When you multiply two powers with the same base, you add the exponents: \(a^m \times a^n = a^{m+n}\).
• Quotient of Powers: When you divide two powers with the same base, you subtract the exponents: \(\frac{a^m}{a^n} = a^{m-n}\).
• Power of a Power: When you raise a power to another power, you multiply the exponents: \((a^m)^n = a^{m \times n}\).

In the original exercise, we used the properties to break down complex exponential expressions into simpler forms. By dividing terms and simplifying, we equated the exponents via properties such as the quotient of powers. This allowed us to solve for the variable effectively.

Properties of Logarithms

Logarithms are the inverse operations of exponents. Knowing how to apply logarithmic properties is essential for solving equations involving logarithms. These are some core properties:

• Log of a Product: \(\log(ab) = \log a + \log b\).
• Log of a Quotient: \(\log(\frac{a}{b}) = \log a - \log b\).
• Log of a Power: \(\log(a^n) = n \log a\).

In the solution, we applied these properties to turn complex multiplicative expressions into additive ones. Specifically, we used the product property to express powers as products of logarithms. This was fundamental in isolating the term involving \(x\) and later solving for \(x \log\left(\frac{b}{a}\right)\).

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying expressions to make them more workable or to solve equations. In the exercise, we often factor, expand, or divide both sides of an equation to achieve this. Key techniques include:

• Factoring: Pulling out a common factor, just like we did with \(2x\), helps to simplify or solve equations more straightforwardly.
• Cancelling Terms: Applying properties like division to simplify expressions; for example, dividing both sides by a term to reduce complexity.
• Substitution: Using equivalent expressions to replace parts of the equation for easier manipulation, like using \(\log\left(\frac{b}{a}\right)\) for \(\log b - \log a\).

In our steps, we simplified the equation step-by-step, using algebraic manipulation to isolate \(x\) and solve the desired expression. This process illustrates the efficiency of using algebra in conjunction with properties of exponents and logarithms.