to simplify this
Log72/75-log50/48+2log25/24
you have to know the rules of logs
I will use:
log (xy) = log x + log y
log(x/y) = log x - log y
log (x^a) = a log x
Log72/75-log50/48+2log25/24
= log [ (72/75) ÷ (50/48) (25/24)^2 ]
= log [ (72/75(48/50)(625/576) ]
= log [1]
= 0
To simplify the expression:
Log72/75 - Log50/48 + 2Log25/24
We can use the logarithmic properties to combine the terms. The two main properties we will use are:
1) Log(x) - Log(y) = Log(x/y)
2) nLog(x) = Log(x^n)
Let's simplify each term step by step:
1) Log72/75 - Log50/48:
Using property 1, we can combine the two terms by subtracting the logarithms of their respective fractions:
Log(72/75 / 50/48)
2) Log(72/75 / 50/48):
To simplify the fraction inside the logarithm, we will divide the numerators and denominators separately:
Log( 72/50 * 48/75 )
3) Log( 72/50 * 48/75 ):
Now, simplify the fraction:
Log( 72/50 * 16/25 )
4) Log( 72/50 * 16/25 ):
Multiply the fractions:
Log( 1152/1250 )
5) Log( 1152/1250 ):
We can simplify the fraction by dividing both the numerator and the denominator by their greatest common factor:
Log( 9/10 )
6) Log( 9/10 ):
Now we can use the property of logarithm (Log(x) - Log(y) = Log(x/y)) to simplify further:
Log( 9 ) - Log( 10 )
7) We can simplify Log( 9 ) and Log( 10 ) as:
Log9 - Log10 = 1 - 1 = 0
Now, let's move on to the next term in the original expression:
8) 2Log25/24:
Using property 2 (nLog(x) = Log(x^n)), we can bring the exponent outside the logarithm:
Log( (25/24)^2 )
9) Log( (25/24)^2 ):
Simplify the expression inside the logarithm:
Log( 625/576 )
10) Log( 625/576 ):
Again, divide the numerator and denominator by their greatest common factor:
Log( 25/23 )
11) Log( 25/23 ):
Using the same property as before, simplify further:
Log( 25 ) - Log( 23 )
12) Finally, we can evaluate the logarithms:
Log( 25 ) - Log( 23 ) = 2 - Log( 23 )
Therefore, the simplified expression is: 2 - Log( 23 )