use properties of logarithms to condense the loarithmic expression. write expression as a sinle logarithm whose coefficient is 1. where possible evaulate the expression.
1/8[3 In(x+5)-In x-In(x^2-4)]
Please show work
1/8[3 In(x+5)-In x-In(x^2-4)]
= (1/8) [ ln (x+5)^3 - lnx - ln (x^2-4) ]
= (1/8) ln ( (x+5)^3 / (x(x^2 - 4) ]
you finish it
What is equal to 435 inches
To condense the logarithmic expression, we can use the properties of logarithms. The properties we will use are:
1) Product Rule: log(base b)(M * N) = log(base b)M + log(base b)N
2) Quotient Rule: log(base b)(M / N) = log(base b)M - log(base b)N
3) Power Rule: log(base b)(M^k) = k * log(base b)M
Given expression: 1/8[3 ln(x+5) - ln(x) - ln(x^2 - 4)]
First, let's apply the product rule to the first term inside the parentheses:
3 ln(x+5) = ln((x+5)^3)
Next, let's apply the quotient rule to the second term:
ln(x) - ln(x^2 - 4) = ln(x / (x^2 - 4))
Since the expression is divided by 8, we can bring the coefficient (1/8) outside the logarithm:
1/8[ln((x+5)^3) - ln(x / (x^2 - 4))]
Now, let's apply the quotient rule to the entire expression:
ln((x+5)^3 / (x / (x^2 - 4)))
Since the coefficient is 1, the final condensed logarithmic expression is:
ln((x+5)^3 / (x / (x^2 - 4)))
To evaluate the expression further, we can simplify the numerator and the denominator:
Numerator:
(x+5)^3 = (x+5)(x+5)(x+5) = (x^2 + 10x + 25)(x+5) = x^3 + 20x^2 + 125x + 125
Denominator:
x / (x^2 - 4) = x / ((x+2)(x-2))
Now, we can substitute the simplified numerator and denominator back into the expression:
ln((x^3 + 20x^2 + 125x + 125) / (x / ((x+2)(x-2))))
This is the final condensed logarithmic expression.