Logarithms question

[seiferseph]

if log2 = x and log3 = y, solve for log(base5)36 in terms of x and y.

how do i even get started on this? i'm really confused with logs.

[Tide]

This seems like a rather roundabout way of doing things. In any case ...

I assume those logarithms are base 10 in which case, by definition of logarithm, we have

10^x = 2
10^y = 3

Let z = \log_5 10 which means 5^z = 10. But

5 = 2 + 3 = 10^x + 10^y

so that

5^z = \left(10^x + 10^y\right)^z = 10

IOW I'm not sure I see the point of the problem!

from which

z \log \left( 10^x + 10^z\right) = \log 10 = 1

and finally

z = \frac {1}{\log \left( 10^x + 10^z \right)}


Of course, a direct approach would have led to

\log_5 10 = \frac {\log 10}{\log_{10} 5} = \frac {1}{\log 5}

[seiferseph]

thanks, but i'm not sure if that is correct (the teacher actually said it was quite simple). the last question was something like a^2 + b^2 for the answer. i'll post a little bit of what i got, i'm not sure if this is right

[Muzza]

Do you realize that your original question is about log_5(10), but the handwritten thing you posted now is about log_5(36)?

[seiferseph]

Do you realize that your original question is about log_5(10), but the handwritten thing you posted now is about log_5(36)?

now i do :blushing: its supposed to be log_5(36)