Problem 146 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

$$ \log _{2} x^{4}+\log _{2} x^{2}=1 $$

Step-by-Step Solution

Answer

The short answer to the question is: The solution for the given logarithmic equation is \(x = \sqrt[6]{2}\).

Step 1: Combine logarithms using the product rule

When adding logarithms with the same base, we can use the product rule, which states: \(\log_b a + \log_b c = \log_b (a * c)\). Apply this rule to the given equation: \(\log _{2} x^{4} + \log _{2} x^{2} = \log _{2} (x^4 * x^2)\)

Step 2: Simplify the product inside the logarithm

Now, we need to simplify the product inside the logarithm. In this case, multiply the two expressions with the same base (x): \(x^4 * x^2 = x^6\) So the equation becomes: \(\log _{2} x^6 = 1\)

Step 3: Remove the logarithm using exponentiation

Since we now have only one logarithm on the left side, we can remove the logarithm by using exponentiation. This basically means raising both sides of the equation to the power of the base of the logarithm: \(2^{\log _{2} x^6} = 2^1\)

Step 4: Simplify both sides of the equation

Simplify the left side of the equation using the property \(b^{\log_b a} = a\): \(x^6 = 2^1\) And simplify the right side of the equation: \(x^6 = 2\)

Step 5: Solve for x

Lastly, solve for x by taking the sixth root of both sides of the equation: \(x = \sqrt[6]{2}\) The solution for the given logarithmic equation is: \(x = \sqrt[6]{2}\)

Key Concepts

Product Rule of Logarithms

The product rule of logarithms is a useful tool when you're faced with adding two logarithms that have the same base. In simpler terms, instead of adding these logs separately, you can combine them into one log. The rule is expressed as:

• \( \log_b a + \log_b c = \log_b (a \times c) \)

By applying this rule, you change two logarithmic expressions into a product inside a single logarithm.
This simplification helps in solving equations, like reducing the number of terms you handle.
In the exercise provided, the rule was used to combine \( \log_2 x^4 \) and \( \log_2 x^2 \) into \( \log_2 (x^4 \times x^2) \). This is an example of how the product rule transforms the expression, making it easier to work with in subsequent steps.

Exponentiation

Exponentiation is the process of turning a logarithm back into a regular number. Think of it as doing the opposite of what the log does. When you have a logarithmic equation like \( \log_2 x^6 = 1 \), exponentiation helps to "undo" the log.

• Here, the base 2 is raised to the power of both sides of the equation.
• This means converting \( \log_2 x^6 \) to \( 2^{\log_2 x^6} \), which simplifies directly to \( x^6 \).

A neat property of logarithms and exponents is that \( b^{\log_b a} = a \). This property allows you to remove the log and directly see what value the expression equals.
In a logarithmic equation, exponentiation efficiently isolates the variable, making it much easier to solve for or simplify as needed.

Simplifying Expressions

Simplifying expressions is a vital skill in algebra, especially when dealing with logarithmic equations. Simplification involves rewriting an expression in a more straightforward or efficient form without changing its value.

• After applying the product rule of logarithms, \( x^4 \times x^2 \) was simplified to \( x^6 \).
• To further simplify, we acknowledge that \( x^6 = 2 \) indicates the need to solve for \( x \) by finding the sixth root of 2.

These steps turn the problem into a less complex form. Simplification cuts down the number of operations needed, allowing for an easier path to the final solution.
With repeated practice, these skills help in quickly identifying patterns and solutions to similar problems in algebra and beyond.