Basic Maths and Logarithm
Logarithmic Equations
JEE Advanced (IIT-JEE 2002)
Grade 11
Question:
The number of positive integers satisfying $x + \log_{10}(2^x + 1) = x \log_{10} 5 + \log_{10} 6$ is:
(1) $0$
(2) $1$
(3) $2$
(4) $\infty$
Step-by-Step Solution
Key Concept: Rewrite: $x(1-\log_{10} 5)+\log_{10}(2^x+1) = \log_{10} 6$. Since $1-\log_{10} 5 = \log_{10} 2$: $\log_{10}(2^x(2^x+1)) = \log_{10} 6$. So $(2^x)^2 + 2^x - 6 = 0 \Rightarrow 2^x = 2$ (reject $2^x = -3$), giving $x = 1$.
The detailed step-by-step mathematical proof is available inside the Mathbee app workspace.
Correct Answer: (2)