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Hello Mr. Sombra Shadow,
I was hoping you could help me with this math question. It is:

Assuming that x, y and z are positive numbers. Use logarithmic properties to write the following expression as a simplified logarithm. The expression is:
2 ln x - 6 ln y + 1/3 ln e^12

Thank you for your assistance.

Answer
There are properties of logarithms that you use to simplify your expression
1)  log x^y = y*log x
2)  log(x/y) = log x - log y
3)  log 10^y = y

The first property says that if you take the log of a number with a power you can pull that exponent off and put it in front of the log as a multiplier
The second property says that the log of a quotient is the difference between the log of the numerator and the log of the denominator
The third property says if the base of the log matches the base of the argument then you get just the exponent

In your problem the 'log' being used is the log base e or ln
So 2 ln x becomes ln x^2 and
-6 ln y becomes - ln y^6
Thus 2 ln x - 6 ln y is by virtue of property 3: ln(x^2/y^6)
And finally (1/3) ln e^12 can be dealt with in two ways:
You can put the (1/3) on the 12 as an exponent e^[12^(1/3)] = e^4 so you have ln e^4 = 4 or
Realize ln e^12 = 12 and then multiply by (1/3) resulting in 4

Professor W

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Sombra Shadow

Expertise

I can answer most questions up through Calculus and some in Number Theory and Abstract Algebra.

Experience

I have had my Bachelor's Degree since 1987 and have been a teacher since 1988. I earned my Masters Degree in Mathematics May 2010. I have been teaching at the same community college since 2002.

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I have taught 12 years at the community college level, medical college, and technical college as well as a high school instructor and alternative education instructor and charter school instructor.

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Master's GPA 3.56 Bachelor's GPA 3.34 Post grad work not degree related GPA 4.0

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