Question

If $1,{\log _9}({3^{1 - x}} + 2),{\log _3}({4.3^x} - 1)$ are in A.P then x is equal to
A. $\log {}_3(4)$
B. $1 - \log {}_3(4)$
C. $1 - \log {}_4(3)$
D. $\log {}_4(3)$

Answer 1

Hint: Before we proceed to solve the problem, it is important to know the definitions of Progression and AP and we should know about the properties of a logarithmic function. We will be using the concept of arithmetic progression and will be reducing it into simpler functions using the required properties.

Formula Used:
${a_n} - {a_{n - 1}} = d$ where d is the common difference.

Complete step by step Solution:
Progressions are numbers arranged in a particular order such that they form an expected order. By expected order, we mean that given some numbers, we can find the next numbers in the series.
Arithmetic Progression (AP) - A sequence is called an arithmetic progression if the difference between any two consecutive terms is always the same. In simple terms, it means that the next number in the series is calculated by adding a fixed number to the previous number in the series. This fixed number is a common difference.

It is given in the question that $1,{\log _9}({3^{1 - x}} + 2),{\log _3}({4.3^x} - 1)$
$ \Rightarrow 2{\log _9}({3^{1 - x}} + 2) = 1 + {\log _3}({4.3^x} - 1)$
$ \Rightarrow {\log _3}({3^{1 - x}} + 2) = {\log _3}3 + {\log _3}({4.3^x} - 1)$
$ \Rightarrow ({3^{1 - x}} + 2) = 3({4.3^x} - 1)$
$ \Rightarrow ({3^{ - x}} + \dfrac{2}{3}) = ({4.3^x} - 1)$
Let ${3^x} = p$
$ \Rightarrow (\dfrac{1}{p} + \dfrac{5}{3}) = 4p$
Rearranging we get,
$ \Rightarrow 12{p^2} - 5p - 3 = 0$
After solving $p = \dfrac{3}{4}$ and $p = - \dfrac{1}{3}$
Since $p = - \dfrac{1}{3}$ is not possible because p cannot be negative.
So, $p = \dfrac{3}{4} = {3^x}$
Now taking a log with base 3 on both sides we get,
$ \Rightarrow x\log 3 = \log (\dfrac{3}{4})$
$ \Rightarrow x = 1 - \log {}_3(4)$

Hence, the correct option is B.

Note: While solving these types of questions we need to know about both progression and logarithmic properties. Care should be taken while applying property $\log ab = \log a + \log b$ and try to relate a, b, and c differently. Also, while dividing by abc, take care that none of them should be zero. If it is zero, we cannot divide.