Logarithm of (x)^m/n with long working out: logarithm of (0.649)^4/3

[Phavonic]

Hi,


I need some help with the process of finding the logarithm of (0.649)^4/3, but using the working out given below.


I get stuck when it comes to using the numerator of the power (this might be a bit hard to write, so I'll avoid the characteristic and mantissa, but hopefully you should see what I'm getting at)


(0.649)^4/3


(6.49 x 10^-1)^4/3


(10^0.8122446968 x 10^-1)^4/3


...this is where I'm not sure what to do with the 4 in the numerator of the power.


If it was to the power 1/3 then I guess I'd just ignore the 1 in the numerator and do..


10^{(-1 + 0.8122446968)/3}


10^{-3 + 2.8122446968/3}


10^(-1 + 0.9374148989)


etc.


So what should I do with that 4 in the numerator of the power in (0.649)^4/3 when doing this process?


Thanks.

 

[Dr.Peterson]

I need some help with the process of finding the logarithm of (0.649)^4/3, but using the working out given below.

I get stuck when it comes to using the numerator of the power (this might be a bit hard to write, so I'll avoid the characteristic and mantissa, but hopefully you should see what I'm getting at)

(0.649)^4/3

(6.49 x 10^-1)^4/3

(10^0.8122446968 x 10^-1)^4/3

...this is where I'm not sure what to do with the 4 in the numerator of the power.

If it was to the power 1/3 then I guess I'd just ignore the 1 in the numerator and do..

10^{(-1 + 0.8122446968)/3}

10^{-3 + 2.8122446968/3}

10^(-1 + 0.9374148989)

etc.

So what should I do with that 4 in the numerator of the power in (0.649)^4/3 when doing this process?

Why can't you continue with the given exponent 4/3?

10^{(-1 + 0.8122446968)*4/3} = 10^{(-3 + 2.8122446968)*4/3} = 10^{-3*4/3 + 2.8122446968*4/3} = 10^{-4 + 2.8122446968*4/3}
​

and so on.

Of course, the quick way is to multiply the log by 4/3:

log(0.649^{4/3}) = 4/3 * log(0.649) = ...
​

 

[lookagain]

I need some help with the process of finding the logarithm of (0.649)^4/3,...

(0.649)^4/3

(6.49 x 10^-1)^4/3

(10^0.8122446968 x 10^-1)^4/3

So what should I do with that 4 in the numerator of the power in (0.649)^4/3 when doing this process?

Phavonic, you must place your fractional exponents inside of grouping symbols because of the Order of Operations:


(0.649)^(4/3)

(0.649)^(4/3)

(6.49 x 10^-1)^(4/3)

(10^0.8122446968 x 10^-1)^(4/3)

(0.649)^(4/3)


This is a separate issue from whether the rest of each expression in each line above is correct.