If $\log_{\sin \frac{\pi}{6}} \left\{\frac{\left|z-2\right| + 3 }{3\left|z - 2\right| - 1 }\right\}>1 $ , then
- $|z - 2| > 7 $
- $|z - 2| < 7 $
- $|z - 2| < 3 $
- $|z - 2| < 6 $
The Correct Option is A
Solution and Explanation
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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)