Question

If \[{\log _{10}}(2)\], \[{\log _{10}}({2^x} - 1)\] and \[{\log _{10}}({2^x} + 3)\] are three consecutive terms of an A.P, then the value of x is
A) \[1\]
B) \[{\log _5}(2)\]
C) \[{\log _2}(5)\]
D) \[{\log _{10}}(5)\]

Answer 1

Hint: In this question we have to find the value of $x$ if three given consecutive terms are in AP. Common difference between the terms are constant in AP use this property to get relation between the terms and then use the property of log to get required value.

Formula used: If a, b, c are the terms in AP then common difference is given as
\[b - a = c - b = d\]

Complete step by step solution: Given: \[{\log _{10}}(2)\], \[{\log _{10}}({2^x} - 1)\] and \[{\log _{10}}({2^x} + 3)\] are three consecutive terms of an A.P
Now in order to find common difference use property of AP
\[{\log _{10}}({2^x} - 1) - {\log _{10}}(2) = {\log _{10}}({2^x} + 3) - {\log _{10}}({2^x} - 1)\]
Use property of log function
\[{\log _{10}}(\frac{{{2^x} - 1}}{2}) = {\log _{10}}(\frac{{{2^x} + 3}}{{{2^x} - 1}})\]
Take inverse log both sides
\[\frac{{{2^x} - 1}}{2} = \frac{{{2^x} + 3}}{{{2^x} - 1}}\]
\[{({2^x} - 1)^2} = 2({2^x} + 3)\]
On simplification we get
\[{2^{2x}} - {2^{x + 1}} + 1 = {2^{x + 1}} + 6\]
\[{2^{2x}} - {2^{x + 2}} = 5\]
Let \[{2^x}\]is equal to z
\[{z^2} - 4z - 5 = 0\]
Use factorization method to get roots of equation
\[{z^2} - 5z + z - 5 = 0\]
\[z(z - 5) + 1(z - 5) = 0\]
\[(z - 5)(z + 1) = 0\]
\[z = - 1\] And \[z = 5\]
Only positive value is valid for log function therefore \[{2^x}\]is equal to positive value
\[{2^x} = 5\]
Now required value is
\[x = {\log _2}(5)\]

Thus, Option (C) is correct.

Note: Use property of AP in these types of questions. Here we must remember that if three consecutive terms are in AP then difference between second term, first term and third term, second term is equal. Log of negative value is not valid.
Sometime students get confused in between AP and GP the only difference in between them is in AP we talk about common difference whereas in GP we talk about common ratio.