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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.

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

Advanced Math

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