Question

What is `log_8(((root5(p))(root(9)q))/(r^2))`?

Answer 1

The solution depends on what the objective is!

One step in the solution takes you to:
`=>1/(log_10(8))[(log_10(p))/5+log_10(q)/9-2log_10(r)]`

The values of `p,q" and "r` are not given.

Explanation:

Demonstrating use of indices in loges.
Source numbers multiplies `->` logs added
Source numbers divided `->` logs subtracted

Conversion of log base into a more usable base
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Consider:`" "root(5)(p) -> p^(1/5)`

Consider:`" "root(9)(q) -> q^(1/9)`

So we have:
`" "log_8((p^(1/5)q^(1/9))/r^2)`
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("Breaking this down into steps")`

`color(blue)("Step 1")`
Divide becomes subtracting logs

`" "log_8((p^(1/5)q^(1/9))/r^2)" "->" "log_8(p^(1/5)q^(1/9))-log_8(r^2)`
'...........................................................

`color(blue)("Step 2")`
Multiply becomes adding logs

`" "[log_8(p^(1/5))+log_8(q^(1/9))]-log_8(r^2)"`

'............................................
`color(blue)("Step 3")`
Source number to a power; multiply the log by the index

`" "[log_8(p)/5+log_8(q)/9]-2log_8(r)"`
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("Step 4")`
Convert the log base, say, base 8 to base 10

Suppose we hade `log_8(a) = (log_10(a))/log_10(8)`

`=> " "[log_10(p)/(5log_10(8))+log_10(q)/(9log_10(8))]-(2log_10(r))/(log_10(8)`

`=>1/(log_10(8))[(log_10(p))/5+log_10(q)/9-2log_10(r)]`

`color(green)("It all depends on where you wish to go from this point")`

By the way: `1/log_10(8)~~1.107` to 3 decimal places