1,log x base y,log y base z,-15log z base x are in AP ..,then??
Useless answer
To determine what happens when the terms -15log z base x, log y base z, and log x base y are in an arithmetic progression (AP), let's first understand the concept of an arithmetic progression.
An arithmetic progression is a sequence of numbers in which the difference between any two consecutive terms is constant. This common difference is denoted by "d."
In this case, the terms are -15log z base x, log y base z, and log x base y. To find the common difference, we need to calculate the difference between consecutive terms.
The difference between the second and first terms:
d1 = log y base z - (-15log z base x) = log y base z + 15log z base x
The difference between the third and second terms:
d2 = (-15log z base x) - log y base z = -15log z base x - log y base z
For the given terms to be in an AP, the two differences (d1 and d2) should be equal.
Therefore, the equation becomes:
log y base z + 15log z base x = -15log z base x - log y base z
Now, let's solve this equation to find the relationship between x, y, and z:
First, let's cancel out the log z base x terms:
log y base z = -2log z base x
Next, let's rewrite the logarithmic equation using exponential form:
y = z^(-2x)
So, the relationship between x, y, and z is y = z^(-2x).
In conclusion, if the terms -15log z base x, log y base z, and log x base y are in an arithmetic progression, then the relationship between x, y, and z is given by the equation y = z^(-2x).
there is a common difference, so
log_y(x) - 1 = log_z(y)-log_y(x)
log_z(y)-log_y(x) = -15log_x(z) - log_z(y)
Not sure just where you want to go with this, since you only have two equations in three unknowns, but if you play around a bit, you should arrive at the fact that both equations work if
x = 1/y = z^3