6ln(x+8)+1/8lnx-ln(cosx)
what about it ?
how do I rewrite it as a single logarithm
6ln(x+8)+1/8lnx-ln(cosx)
= ln[(x+8)^6] - ln(x^(1/8)) - ln(cosx)
= ln [(x+8)^6/(x^(1/8)cosx)
The given expression is:
6ln(x+8) + 1/8ln(x) - ln(cos(x))
To simplify this expression, we can use some properties of logarithms.
1. The logarithm of a product is equal to the sum of the logarithms of the individual factors.
2. The logarithm of a quotient is equal to the difference of the logarithms of the numerator and denominator.
Using these properties, let's simplify the expression step by step:
Step 1: Simplify the first term, 6ln(x+8):
6ln(x+8) = ln((x+8)^6)
Step 2: Simplify the second term, 1/8ln(x):
1/8ln(x) = ln(x^(1/8))
Step 3: Simplify the expression further:
ln((x+8)^6) + ln(x^(1/8)) - ln(cos(x))
Since the logarithms have the same base (natural logarithm), we can combine them into a single logarithm using the addition and subtraction properties:
ln((x+8)^6 * x^(1/8)) - ln(cos(x))
Step 4: Simplify the combined logarithm:
ln(((x+8)^6 * x^(1/8)) / cos(x))
And that's the simplified form of the given expression.