Question

If \[{\log _{10}}2,{\log _{10}}\left( {{2^x} - 1} \right),{\log _{10}}\left( {{2^x} + 3} \right)\] are three consecutive terms of an AP, then which one of the following is correct
A. \[x = 0\]
B. \[x = 1\]
C. \[x = {\log _2}5\]
D. \[x = {\log _5}2\]

Answer 1

Hint: In our case, we have been given the three consecutive terms of Arithmetic progression such as \[{\log _{10}}2,{\log _{10}}\left( {{2^x} - 1} \right),{\log _{10}}\left( {{2^x} + 3} \right)\] and now we have to determine the correct condition as per the given statement. For that we have to write the given terms with respect to \[2b = a + c\] and then substituting the given terms in \[2b = a + c\] and solve accordingly using power rule of log to get the desired solution.

Formula Used: Product rule:
\[{\log _{\rm{a}}}({\rm{mn}}) = {\log _{\rm{a}}}{\rm{m}} + {\log _{\rm{a}}}{\rm{n}}\]
Quotient rule:
\[{\log _{\rm{a}}}\dfrac{{\rm{m}}}{{\rm{n}}} = {\log _{\rm{a}}}{\rm{m}} - {\log _{\rm{a}}}{\rm{n}}\]
Power rule:
\[{\log _{\rm{a}}}{{\rm{m}}^{\rm{n}}} = {\rm{n}}{\log _{\rm{a}}}{\rm{m}}\]

Complete step by step solution: We have been provided in the question that,
The A.P's three consecutive terms are
\[{\log _{10}}2,{\log _{10}}\left( {{2^x} - 1} \right),{\log _{10}}\left( {{2^x} + 3} \right)\]
And we are asked to determine the condition that is correct according to the given statement.
We have been already known that if \[a,b,c\] are in A.P then
\[2b = a + c\]
Now, we have to wrote the given terms in terms of the above formula, we obtain
\[ \Rightarrow 2 \times {\log _{10}}\left( {{2^x} - 1} \right) = {\log _{10}}2 + {\log _{10}}\left( {{2^x} + 3} \right)\]
Now, using power rule (\[{\log _{\rm{a}}}{{\rm{m}}^{\rm{n}}} = {\rm{n}}{\log _{\rm{a}}}{\rm{m}}\]) we can write the above equation as
\[ \Rightarrow {\log _{10}}{\left( {{2^x} - 1} \right)^2} = {\log _{10}}\left( {2 \times \left( {{2^x} + 3} \right)} \right)\]
Now, after neglecting log, we have
\[ \Rightarrow {\left( {{2^x} - 1} \right)^2} = 2 \times \left( {{2^x} + 3} \right)\]---- (1)
Now, for instance let us assume
\[{2^x} = t\]
Now, on rewriting the equation (1) in terms of above assumption \[{2^x} = t\] we get
\[ \Rightarrow {\left( {t - 1} \right)^2} = 2 \times \left( {t + 3} \right)\]
Now using \[{\left( {a - b} \right)^2}\] formula, expand the left side of the above equation we get
\[ \Rightarrow {t^2} - 2t + 1 = 2(t + 3)\]
Now, we have to solve the right side of the equation by multiplying 2 with the terms inside the parentheses we have
\[ \Rightarrow {t^2} - 2t + 1 = 2t + 6\]
Now, we have to group the like terms to simplify, we get
\[ \Rightarrow {t^2} - 4t - 5 = 0\]
Now, we have to write the above expression in terms of factors, we obtain
\[ \Rightarrow (t - 5)(t + 1) = 0\]
Now, we have to set factors equal to zero, we get
\[t + 1 = 0\;\]OR \[t - 5 = 0\]
On soling for t, we get
\[t = - 1\] OR \[t = 5\]
Since, \[{2^x} > 0\] now it implies,
\[{2^x} \ne - 1\]
Now, we have that
\[{2^x} = 5\]
Now, we have to write in terms of log for that take log on both sides, we get
\[{\log _2}x = {\log _2}5\]
Now, rewrite the above expression using the property\[\log {a^b} = b\log a\] we have
\[x{\log _2}2 = {\log _2}5\]
We know that \[{\log _2}2 = 1\] so, we have
\[x = {\log _2}5\]
Therefore, if \[{\log _{10}}2,{\log _{10}}\left( {{2^x} - 1} \right),{\log _{10}}\left( {{2^x} + 3} \right)\] are three consecutive terms of an AP, then \[x = {\log _2}5\] is correct

Option ‘C’ is correct

Note: As this problem has more formulas to be remembered, we have to solve these types of problems carefully. Students should be thorough with properties of logarithm and log rules like quotient rule, product rule and power rule in order to get the correct solution. Because applying wrong formula in wrong step leads to wrong solution.