Question

If $x = {\log _b}a$, $y = {\log _c}b$ and $z = {\log _a}c$, then find the value of $xyz$.
A. $0$
B. $1$
C. $2$
D. $4$

Answer 1

Hint: First we will rewrite each value of x, y, z with base 10. Then multiply each value to get the value of $xyz$.

Formula Used:
${\log _n}m = \dfrac{{\log m}}{{\log n}}$

Complete step by step solution:
Given that
$x = {\log _b}a$
$y = {\log _c}b$
$z = {\log _a}c$
Here it is asked to find the value of xyz.
We know that
${\log _n}m = \dfrac{{\log m}}{{\log n}}$
That is
$x = {\log _b}a$
$x = \dfrac{{\log a}}{{\log b}}$. . . . . (1)
$y = {\log _c}b$
$y = \dfrac{{\log b}}{{\log c}}$. . . . . (2)
$z = {\log _a}c$
$z = \dfrac{{\log c}}{{\log a}}$. . . . . (3)
Hence the required value can be found as
$xyz = \dfrac{{\log a}}{{\log b}} \times \dfrac{{\log b}}{{\log c}} \times \dfrac{{\log c}}{{\log a}}$. . . . . (from equations (1),(2),(3))
Cancelling out the common terms, we get
$xyz = 1$

Option ‘B’ is correct

Additional information:
There are two types of logarithms. The two types of logarithms are natural logarithm and common logarithm.
The base of the natural logarithm is e. e is known as Euler’s constant.
The base of common logarithm is 10.
There is some property:
1) Power Rule:
${\log _a}{m^n} = n{\log _a}m$
2) Product rule:
${\log _a}\left( {mn} \right) = {\log _a}m + {\log _a}n$
3) Quotient rule:
 ${\log _a}\left( {\dfrac{m}{n}} \right) = {\log _a}m - {\log _a}n$
4) Zero rule:
${\log _a}1 = 0$

Note: Students can get confused while writing the given terms into fractions. Always remember that logarithm is the inverse of the exponential. Start by determining the base, which is represented in the equation by b, the exponent which is y, and the exponential expression which is x in order to solve a logarithm.