Problem 12 - Comprehensive Trigonometry for IIT JEE Main and Advanced Rejaul Makshud MeGraw Hill

Question

If \(\log _{10} 2=x\), then \(\log _{10} 5\) is (a) 1 (b) \(1-x\) (c) \(x+1\) (d) \(2 x\)

Step-by-Step Solution

Answer

None of the given options a, b, c, and d are correct

Step 1: Understand the given

We're given that \( \log_{10} 2 = x \). We need to determine the value of \( \log_{10} 5 \) using the values provided in options (a) through (d).

Step 2: Apply properties of logarithms

We know that \( \log_b a = \frac{1}{\log_a b} \). So, we can rewrite \( \log_{10} 5 \) as \( \frac{1}{\log_5 10} \).

Step 3: Apply properties of numbers

We can further simplify by knowing that \( \log_a b = c \) implies \( a^c = b \) and vice-versa. So, we know \( 10 = 2^x \times 5^x \). Therefore, we can rewrite \( \log_{10} 5 \) as \( \frac{1}{x} \).

Step 4: Verify with options

Comparing the simplified expression with the given options shows us that the value of \( \log_{10} 5 \) is not equal to any of the provided options, thus none of the options (a) through (d) are correct in this context.

Key Concepts

Properties of Logarithms

Logarithms have several important properties that simplify complex expressions, making it easier to solve logarithmic equations. To grasp how they work, think back to exponents, as logarithms are essentially their inverse. Here are some of the fundamental properties:

• Product Property: The logarithm of a product equals the sum of the logarithms: \( \log_b (MN) = \log_b M + \log_b N \).
• Quotient Property: The logarithm of a quotient equals the difference of the logarithms: \( \log_b \left(\frac{M}{N}\right) = \log_b M - \log_b N \).
• Power Property: The logarithm of a power equals the exponent times the logarithm: \( \log_b (M^p) = p \cdot \log_b M \).

Understanding these properties allows us to manipulate and simplify logarithmic expressions, such as rewriting \( \log_{10} 5 \) using properties to potentially find its relation to given variables.

Logarithmic Equations

Solving logarithmic equations involves applying the properties of logarithms to isolate the unknown variable. Start by equating the equation's base. Then use properties to transform the equation into a solvable form.In the provided exercise, knowing that \( \log_{10} 2 = x \) can help us express other logarithms in terms of \( x \). For instance, you could initially look for relationships between the given terms by examining common bases or using exponential transformations, as done to deduce \( 10 = 2^x \times 5^x \). This insight allows us to experiment with expressions involving \( x \) to identify equivalent values for \( \log_{10} 5 \).Often, solving such equations might involve reversing the conversion by moving between logarithmic and exponential forms. Be sure to check your solutions against the problem constraints to ensure accuracy.

Logarithmic Identities

Logarithmic identities help us in expressing complex logarithmic statements in simpler forms and are particularly useful for solving and proving equations. Some key identities are:

• Change of Base Formula: This allows the conversion of a logarithm to another base: \( \log_b a = \frac{\log_c a}{\log_c b} \).
• Identity Logarithm: \( \log_b b = 1 \), because any number to the power of 1 is itself.
• Zero Logarithm: \( \log_b 1 = 0 \), since any base raised to the power 0 equals 1.

Understanding these core identities allows us to manipulate expressions efficiently. For instance, using the change of base, you can transform \( \log_{10} 5 \) into another base to explore possible relationships or compare it against known expressions related to \( x \). This is particularly useful when the solutions do not initially match given options, prompting deeper analysis or verification in the context of logarithmic equations.