SOLUTION: show that {{2log(16/15)+log(25/24)-log(32/27)}}=0

Question 595609: show that
{{2log(16/15)+log(25/24)-log(32/27)}}=0
Answer by jsmallt9(3758) (Show Source): You can put this solution on YOUR website!

To find out if the left side of the equation is really equal to zero we need to combine/condense them down to a single logarithm. And since the arguments of the logarithms are different, we will need to use the following properties of logs:

These properties require that the coefficients of the logs be 1's. Fortunately there is another property of logarithms, , which allows us to "move" a coefficient into the argument as its exponent.

So we will start by using the third property to move the 2 in front of the first log:

Now we can start combining terms. The first two logs have a "+" between them so we will use the first property (which also has a "+" between the logs):

The remaining logs have a "-" between them so we will use the second property (which also has a "-" between the logs):

Now we will set about to simplify the big fraction inside the log. Since dividing is the same as multiplying by a reciprocal, I'm going to rewrite the "divide by 32/27" into "multiply by 27/32":

Next I'm going to start canceling factors that are common to the numerators and denominators. In order to see all the factors I am going to first rewrite the squared fraction without an exponent:

and then I'll factor the numerators and denominators:

Now we can start canceling:

As you can see, everything cancels out! So all we are left with is:

And since the zero power of 10 is 1, log(1) is 0:
0 = 0 Check!

Note: The bases of the logarithms could have been any number (as long as all three bases were the same) since the zero power of any base is equal to 1!
RELATED QUESTIONS
show that... (answered by jvdiwate@rediffmail.com)

Prove that... (answered by Boreal)

simplify: log 2+16 log(16/15)+12 log(25/24)+7... (answered by jsmallt9)

Simplify,... (answered by stanbon)

Log(75/16)-2log(5/9)+log(32/243) (answered by jsmallt9)

log(75/16)+2log(5/4)+log(32/243)=log2 (answered by MathLover1,Alan3354)

log 16/log... (answered by jsmallt9)

LOG 16/LOG... (answered by MathLover1)

Solve the equation 3log[base5]x - log[base5]4 = log[base5]16. Solve the equation... (answered by drk)