A log question??

Question

If log (base 3) x = log (base 5) y, find in the form of a single logarithm, the value of log (base x) y.
Please include the steps as well.

Answer 1

log (base 3) x = log (base 5) y

Note, log(base 10) x can be written as lg x.
So I convert log (base 3) x to (lg x)/(lg 3) and similarly
log (base 5) y to (lg y)/(lg 5)

Hence,
log (base 3) x = log (base 5) y
(lg x)/(lg 3) = (lg y)/(lg 5)
(lg x)/[(lg 3)(lg y)] = 1/(lg 5)
(lg x)/(lg y) = (lg 3)/(lg 5)

To find log (base x) y or converted to the lg form, (lg y)/ (lg x),
Therefore,
(lg y)/ (lg x) = 1/[(lg 3)/(lg 5)]
= (lg 5)/(lg 3)
= 1.46 (3 sf)

Pts to note though are the rules of logarithms.
To simplify things, I'll use log(base 10), but the rules 1 and 2 below apply to log(any base).

1) lg a * lg b = lg (a+b)

2) (lg a)/(lg b) = lg (a-b)

3) log (base a) x = y
to convert to exponential form,
x = y^a (y raised to the power of a)

4) To convert log (base a) x to the form of lg or log(base 10)
log (base) number = [lg ( of the number) / lg (of the base)]

eg. log (base 3) x = (lg x) / (lg 3)

Hope this helps.

Answer 2

Call a = log(base3)x so x=3^a
Call b= log (base5)y so y=5^b As a= b x=3^a and y=5^a
If you call z = log(base x)y this means that x^z=y

so 3^az = 5^a so (3^z)^a= 5^a so [3^z/5]^a =1

As a is not zero because x would be 1 and the base can´t be 1 3^z/5 =1 so 3^z= 5 and z= log(base3)5

Answer 3

log (base 3) x = log (base 5) y
if z = log (base 3) (x), then
3^z = x
5^z = y
log (base x) (y) = log(base x) (5^z)
log (base x) (y) = z*log(base x) (5)
log (base x) (y) = (log (base 3) (x))(log(base x) (5))