Write this expression as a single logarithm
6ln(x+8)+1/8lnx-ln(cosx)
Use the product rule,
ln(a)+ln(b) = ln(ab)
ln(a)-ln(b) = ln(a/b)
Work out the problem and post your answer for a check if you wish.
Also, see Reiny's answer to your previous post.
http://www.jiskha.com/display.cgi?id=1277479844
To write the given expression as a single logarithm, we can use the properties of logarithms. There are three main properties we will utilize:
1. Product Rule: log(a) + log(b) = log(ab)
2. Quotient Rule: log(a) - log(b) = log(a/b)
3. Power Rule: log(a^b) = b * log(a)
Now, let's simplify the given expression step by step:
1. 6ln(x+8) + 1/8ln(x) - ln(cosx)
Applying the Product Rule to the first term: 6ln(x+8) = ln((x+8)^6)
The expression is now: ln((x+8)^6) + 1/8ln(x) - ln(cosx)
2. Applying the Quotient Rule to the second term: 1/8ln(x) = ln(x^(1/8))
The expression becomes: ln((x+8)^6) + ln(x^(1/8)) - ln(cosx)
3. Applying the Power Rule to the second term: ln(x^(1/8)) = (1/8) * ln(x)
We now have: ln((x+8)^6) + (1/8) * ln(x) - ln(cosx)
4. We can now apply the Product Rule again to combine the three terms: ln((x+8)^6 * x^(1/8) / cosx)
Therefore, the expression can be written as a single logarithm:
ln((x+8)^6 * x^(1/8) / cosx)