Question

If \( y = a + b \log_e x \) then which of the following is true?

• A. \( \log_e y \) is proportional to x
• B. \( e^y \) is proportional to \( x^b \)
• C. \( y - a \) is proportional to \( x^b \)
• D. \( \frac{1}{y-a} \) is proportional to \( x^b \)

Hint:

When checking proportionality statements involving logs and exponents, try to isolate the terms mentioned in the option. If an option involves \(e^y\), your goal is to manipulate the original equation to get \(e^y\) on one side.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
This question tests the understanding of proportionality and the manipulation of logarithmic and exponential functions. A quantity Y is proportional to a quantity X if \( Y = kX \) for some non-zero constant k.
Step 2: Key Formula or Approach:
We need to manipulate the given equation \( y = a + b \log_e x \) algebraically to match one of the given proportionality statements. The key properties to use are:

\( k \log x = \log(x^k) \)
\( e^{\log_e x} = x \)
\( e^{u+v} = e^u e^v \)
Step 3: Detailed Explanation:
Let's start with the given equation: \[ y = a + b \log_e x \] We want to test the statement "\( e^y \) is proportional to \( x^b \)". This means we need to see if we can arrive at an equation of the form \( e^y = (\text{constant}) \times x^b \). Let's rearrange the initial equation to isolate the logarithmic term: \[ y - a = b \log_e x \] Using the power rule of logarithms: \[ y - a = \log_e(x^b) \] Now, to get \( e^y \), we should exponentiate both sides with base e: \[ e^{y-a} = e^{\log_e(x^b)} \] Using the property that exponentiation and logarithms are inverse functions: \[ e^{y-a} = x^b \] Using the property of exponents \( e^{u-v} = e^u / e^v \): \[ \frac{e^y}{e^a} = x^b \] Multiplying both sides by \( e^a \): \[ e^y = e^a \cdot x^b \] Since 'a' is a constant, \( e^a \) is also a constant. Let's call it \( k = e^a \). Then we have: \[ e^y = k \cdot x^b \] This is the definition of proportionality. Therefore, \( e^y \) is proportional to \( x^b \).
Step 4: Final Answer:
The statement "\( e^y \) is proportional to \( x^b \)" is true.