SOLUTION: prove that if a series of numbers are in GP,their logarithms are in AP

Question 970739: prove that if a series of numbers are in GP,their logarithms are in AP
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
geometric progression is An = A1 * r^(n-1)

arithmetric progression is An = A1 + (n-1)*d

start with the geometric progression of An = A1 * r^(n-1)

take the log of both sides of that equation to get:

log(An) = log(A1 * r^(n-1))

since log(a*b) = log(a) + log(b), this equation becomes:

log(An) = log(A1) + log(r^(n-1))

since log(a^b) = b*log(a), this equation becomes:

log(An) = log(A1) + (n-1)*log(r)

that's an arithmetic rogression with the common difference equal to log(r).

we'll take an example:

A1 = 100
r = 1.5
n = 5

geometric progression:

An = A1 * (r^(n-1) which becomes:
A5 = 100 * 1.5^4 which becomes:
A5 = 506.25

now we want to find log(A5).

if we are correct, then log(A5) should be equal to log(506.25).

start with:

log(A1) = log(100)
r = 1.5
n = 5

use the arithmetic progression of:

log(A5) = log(A1) + (n-1)*log(r) which becomes:
log(A5) = log(100) + 4*log(1.5) which becomes:
log(A5) = log(100) + log(1.5^4) whcih becomes:
log(A5) = log(100*1.5^4) which becomes:
log(A5) = log(506.25)

A5 = 506.25
log(A5) = log(506.25)

looks like we're good.

RELATED QUESTIONS
If 1/y+ax,1/2y,1/y+az are in AP prove that a^2 x,y,z are in... (answered by richwmiller)

a^(1/x)=b^(1/y)=c^(1/z) if a,b,c are in GP. Prove x,y,z are in... (answered by jsmallt9)

three numbers are in AP and their sum is 21. if 1,5,15 are added to them... (answered by Fombitz)

find the three digit number whose digits are in GP and and the digits of a number that is (answered by Edwin McCravy)

if a,b,c in ap nd x,y,z in gp prove that x^b y^c z^a=x^c y^a... (answered by fractalier)

if a,8,b are in AP and a,4,b are in GP a,x,b are in HP then x... (answered by richwmiller)

if a,b,c are in AP, aalnd a,mb,c are in GP then a, m^2b, c are in... (answered by MathLover1)