Question

: If \[x,y,z\] are in GP and \[{a^x} = {b^y} = {c^z}\] then
A.\[{\log _a}c = {\log _b}a\]
B.\[{\log _b}a = {\log _c}b\]
C.\[{\log _c}b = {\log _a}c\]
D.None of these

Answer 1

Hint: The Logarithmic Functions have some of the properties that allow you to simplify the logarithms when the input is in the form of product, quotient or the value taken to the power. Some of the properties are listed below:
Product Rule: \[\log ab = \log a + \log b\]
Quotient Rule: \[\log \dfrac{a}{b} = \log a - \log b\]
Power Rule : \[a\log x = \log {x^a}\]
Change of Base Rule: \[{\log _a}b = \dfrac{{\log b}}{{\log a}}\]

Complete step-by-step answer:
A geometric progression or a geometric sequence is the one, in which each term is varied by another by a common ratio. General form of a GP is
\[a,ar,a{r^2},a{r^3},...,a{r^n}\]
Where \[a\] is the first term
\[r\] Is the common ratio
\[a{r^n}\] is the last term
If \[a,b,c\] are in GP then \[{a^2} = bc\] .
We are given that \[x,y,z\] are in GP.
Therefore we have \[{y^2} = xz\] … (1)
We are also given that \[{a^x} = {b^y} = {c^z}\]
Taking \[\log \] we get the following:
\[x\log a = y\log b = z\log c = k\]
Therefore we get
\[x = \dfrac{k}{{\log a}},y = \dfrac{k}{{\log b}},z = \dfrac{k}{{\log c}}\]
Putting these values in equation (1) we get
\[\dfrac{{{k^2}}}{{{{\left( {\log b} \right)}^2}}} = \dfrac{{{k^2}}}{{\left( {\log a} \right)\left( {\log c} \right)}}\]
On simplification we get
\[\dfrac{{\log a}}{{\log b}} = \dfrac{{\log b}}{{\log c}}\]
On using the logarithmic property we get
\[{\log _b}a = {\log _c}b\]
Therefore option (B) is the correct answer.
So, the correct answer is “Option B”.

Note: A Geometric Progression (GP) is a type of sequence where each succeeding term is produced by multiplying each preceding term by a fixed number, which is called a common ratio. This progression is also known as a geometric sequence of numbers as it follows a pattern.