An in-depth guide on the relationship between logarithmic and exponential functions as inverse operations for CBSE Class 11 Applied Mathematics, covering theory and step-by-step solutions.
Logarithmic and exponential functions are inverse operations. An exponential function expresses growth, while a logarithmic function expresses the exponent required to reach a specific value. If the exponential form is b^x = y, the equivalent logarithmic form is log_b(y) = x, where b > 0 and b ≠ 1.
• Identify the base (b = 2), the exponent (x = 5), and the result (y = 32).
• Apply the definition log_b(y) = x.
• Substitute the identified values: log_2(32) = 5.
Answer: log_2(32) = 5
Since these functions are inverses, they 'undo' each other. Applying a logarithm to an exponential expression with the same base simplifies to the exponent, and raising a base to a logarithm with the same base simplifies to the argument.
• Recognize the property b^{log_b(x)} = x.
• Identify the base b = 5 and the argument x = 25.
• Apply the identity directly.
Answer: 25
To solve an equation where the variable is an exponent, convert the exponential equation to logarithmic form. This allows us to isolate the variable using logarithmic properties.
• Write the equation in logarithmic form: x = log_4(64).
• Express 64 as a power of 4, i.e., 64 = 4^3.
• Use the property log_b(b^n) = n.
• Therefore, x = log_4(4^3) = 3.
Answer: x = 3