SOLUTION: I'm completely done with my assignment, But i'm on my last question and it's a critical thinking question. It says, 2(log 2x - log y)-(log 3 + 2log 5). I got told the answer is: lo

Question 549397: I'm completely done with my assignment, But i'm on my last question and it's a critical thinking question. It says, 2(log 2x - log y)-(log 3 + 2log 5). I got told the answer is: log 4x^2/75y^2 but i don't know how to work it out and explain my work. It's a difficult question and if you could help me that would be great. Thanks for your help.
Found 3 solutions by KMST, nerdybill, bucky:
Answer by KMST(5328)   (Show Source): You can put this solution on YOUR website!

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Answer by nerdybill(7384)   (Show Source): You can put this solution on YOUR website!
I'm completely done with my assignment, But i'm on my last question and it's a critical thinking question. It says, 2(log 2x - log y)-(log 3 + 2log 5). I got told the answer is: log 4x^2/75y^2 but i don't know how to work it out and explain my work. It's a difficult question and if you could help me that would be great. Thanks for your help.
You need to apply "log rules":
2(log 2x - log y)-(log 3 + 2log 5)
distribute the 2 and remove parentheses:
2log 2x - 2log y-log 3 - 2log 5
multiplier of a log can be thought of as an exponent:
log(2x)^2 - logy^2 - log(3) - log 5^2
subtraction can be thought of as division:
log(4x^2) - logy^2 - log(3) - log 25
log(4x^2)/y^2 - log(3) - log 25
log(4x^2)/3y^2 - log 25
log(4x^2)/(75y^2)

Answer by bucky(2189)   (Show Source): You can put this solution on YOUR website!
Given to simplify:
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Look at the first term, specifically look inside the parentheses. The parentheses contain the difference of two logarithms and by the rules governing logarithms, the difference of the logarithms of two quantities is equal to the logarithm of the division as shown in the following:
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Next by the rules of logarithms, the 2 that multiplies the logarithm of the simplified first term can become the exponent of the quantity that the logarithm is operating on. When this rule is applied, the first term changes as shown in the following expression:
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Still operating on the first term, use the power rule of exponents to square the quantity that the log function is operating on and get:
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Moving on to the second term in the expression, change the 2 that multiplies log(5) so that it becomes the exponent of 5 as in the following:
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and square the 5 to get:
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Now recognize that the sum of two logarithms equals the log of the product of the quantities that the two logarithms are operating on:
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Multiply out the 3*25 to get 75 so that the expression now is:
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You again have the difference of two logarithms which means that by the rules of logarithms you can change this to the logarithm of the division of the two quantities that the logarithms are operating on:
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Dividing by 75 is the same as multiplying by 1/75 and this gives you the final simplified answer of:
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Hope this helps you to understand the problem. Make sure you understand each of the rules for logarithms that are applied in solving this problem.
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