# Log5(x) = 16logx(5) solve for x

log5(x) = 16logx(5)
solve for x.

With this one, I have no idea where to start. All I have even thought about doing is bringing up the 16 to make it 5^16 but that doesn't seem to help me.

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Mark44
Mentor

Convert one of your logs so that both logs are in the same base. Do you have a formula for converting from one log base to another?

No i dont have a formula to do that

the formula is:

log5(x) = log (x) / log (5)

Mark44
Mentor

Let y = logbx
Then x = by
So log x = log(by) = y log b
And y = (log x)/(log b)

Hence logbx = (log x)/(log b)

In the third step above, you can use any log base you want. I used the common log (log10).

ok thnx for the formulas, ok so i have the log base for one side which is (log10(x)) / (log10(5)) what do i do now?

Mark44
Mentor

Replace log5(x) in your original equation.

When you do that, what does your equation become?

(log10(x)) / (log10(5)) = logx(5^16)

Mark44
Mentor

ok thnx for the formulas, ok so i have the log base for one side which is (log10(x)) / (log10(5)) what do i do now?
Instead of changing log5 to log, why don't you change logx to log5? The goal is to be using the same log base on both sides of the equation.

so are you saying change it so it is: log5(5^16) / log5(x) = log5(x) ????

Mark44
Mentor

Yes. Now put it in the context of the original equation.

log5x = logx516
==> log5x = [log5516]/log5x

The numerator on the right can be simplified to just plain 16, and you can multiply both sides by log5x.

omg, thankyou so much! i get it finally. I wish i could think like you

Mark44
Mentor

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