The sum of all the real roots of the equation \((e^{2x} – 4)(6e^{2x} – 5e^x + 1) = 0\) is
- \(log\;e^3\)
- \(–log\;e^3\)
- \(log\;e^6\)
- \(–log\;e^6\)
The Correct Option is B
Solution and Explanation
Given equation : \((e^{2x} – 4)(6e^{2x} – 5e^x + 1) = 0\)
\(⇒ e^{2x} – 4 = 0 \;or \;6e^{2x} – 5e^x + 1 = 0\)
\(⇒ e^{2x} = 4 \;or\; 6(e^x)^2 – 3e^x – 2e^x + 1 = 0\)
\(⇒ 2x = ln4 \;or (3e^x – 1)(2e^x – 1) = 0\)
\(⇒ x = In2 \;or\; e^x = \frac{1}{3}\; or\; e^x = \frac{1}{2}\)
else, \(x = ln\frac{1}{3}, -ln2\)
Sum of all real roots = \(ln2 – ln3 – ln2\)
= \(–ln3\)
Hence, the correct option is (B): \(–log\;e^3\)
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Concepts Used:
Exponential and Logarithmic Functions
Logarithmic Functions:
The inverses of exponential functions are the logarithmic functions. The exponential function is y = ax and its inverse is x = ay. The logarithmic function y = logax is derived as the equivalent to the exponential equation x = ay. y = logax only under the following conditions: x = ay, (where, a > 0, and a≠1). In totality, it is called the logarithmic function with base a.
The domain of a logarithmic function is real numbers greater than 0, and the range is real numbers. The graph of y = logax is symmetrical to the graph of y = ax w.r.t. the line y = x. This relationship is true for any of the exponential functions and their inverse.
Exponential Functions:
Exponential functions have the formation as:
f(x)=bx
where,
b = the base
x = the exponent (or power)