Use properties of logarithms to condense the logarithmic expression. Write the expression as a single logarithm whose coefficient is 1. Where possible, evaluate logarithmic expressions.
8 ln x -(1/3) ln y
Choose one answer.
a. ln x^8 y^3
b. ln x^8 /3"sqrt"y
c. ln x^8 3"sqrt"y
d. ln x^8/y^3
The answer I originally came up with, was ln(x^8y^3), but this is not an option, since answer a does not have parenthesis.
8 ln x -(1/3) ln y
= ln(x^8) - ln(y^(1/3))
= ln (x^8/(y^(1/3))
if in b)
you mean ln(x^8/cuberoot(y))
that would be it, and the same as mine
To condense the logarithmic expression 8 ln x - (1/3) ln y, you can use the property of logarithms which states that the difference of logarithms is equal to the logarithm of the quotient of the corresponding arguments.
Applying this property, you can rewrite the expression as a single logarithm with coefficient 1:
8 ln x - (1/3) ln y = ln(x^8) - ln(y^(1/3))
Now, using another property of logarithms, the difference of logarithms is equal to the logarithm of the quotient of the arguments:
ln(x^8) - ln(y^(1/3)) = ln(x^8 / y^(1/3))
Finally, using the properties of exponents, you can simplify the expression further:
ln(x^8 / y^(1/3)) = ln(x^8 / (y^(1/3))^3) = ln(x^8 / y)
Therefore, the condensed expression is ln(x^8 / y).
Note: None of the given options match the condensed expression exactly, but the closest one is option d. ln(x^8/y^3).
To condense the given logarithmic expression, you can use the properties of logarithms.
The property you need to use in this case is the power rule of logarithms, which states that log(base a)(b^n) = n * log(base a)(b).
Applying this property to the given expression, you can rewrite it as:
8 ln x - (1/3) ln y
Using the power rule, the expression becomes:
ln(x^8) - ln(y^(1/3))
Next, you can use another property of logarithms, which is the subtraction rule, stating that log(base a)(b) - log(base a)(c) = log(base a)(b/c).
Applying this property, you can combine the two logarithms:
ln(x^8 / y^(1/3))
Simplifying further, you can rewrite the expression as a single logarithm:
ln((x^8) / (y^(1/3)))
However, none of the given answer options match this result exactly. To find the correct answer, you can evaluate the expression further.
To evaluate the expression, you can convert the exponent of 1/3 to a fractional exponent and use the power rule of exponents.
The expression can be rewritten as:
ln(x^8 / (y^(1/3)))
= ln(x^8 / (y^(1/3))^3)
= ln(x^8 / y)
So, the correct condensed form of the expression is ln(x^8 / y), which corresponds to answer option (d).