chande into a single logarithm
(1/2)ln(x^(2)+1)-4ln(1/2)-((1/2)[ln(x-4)+lnx] )
steps to please
To combine the given expression into a single logarithm, let's break it down step by step:
Step 1:
Using the properties of logarithms, we can simplify the given expression by applying the rules of logarithmic arithmetic. The first step is to simplify the expression inside each logarithm.
(1/2)ln(x^2 + 1) - 4ln(1/2) - ((1/2)(ln(x - 4) + lnx))
Step 2:
Using the power rule of logarithms, we can write x^2 as (x^2 + 1) - 1.
(1/2)ln((x^2 + 1) - 1) - 4ln(1/2) - ((1/2)(ln(x - 4) + lnx))
Step 3:
Applying the product rule of logarithms, we can combine the two logarithms inside the parentheses.
(1/2)ln((x^2 + 1) - 1) - 4ln(1/2) - ((1/2)ln((x - 4)x))
Step 4:
Using the quotient rule of logarithms, we can combine the two logarithms at the end of the expression.
(1/2)ln((x^2 + 1) - 1) - 4ln(1/2) - (ln((x - 4)x)^(1/2))
Step 5:
Using the identity ln(a) - ln(b) = ln(a/b), we can simplify the first two terms.
(1/2)ln(x^2) - 4ln(1/2) - (ln((x - 4)x)^(1/2))
Step 6:
Using the power rule of logarithms, we can simplify the square root of (x - 4)x.
(1/2)ln(x^2) - 4ln(1/2) - (1/2)ln((x - 4)x)
Step 7:
Using the logarithmic rule ln(a) - ln(b) = ln(a/b), we can simplify the terms involving ln(1/2).
(1/2)ln(x^2) - ln((1/2)^4) - (1/2)ln((x - 4)x)
Step 8:
Evaluate (1/2)^4, which simplifies to 1/16.
(1/2)ln(x^2) - ln(1/16) - (1/2)ln((x - 4)x)
Step 9:
Simplify ln(1/16) using the logarithmic rule ln(a/b) = ln(a) - ln(b).
(1/2)ln(x^2) - (-4ln(2)) - (1/2)ln((x - 4)x)
Step 10:
Multiply -4ln(2) by (-1/2) to simplify the expression.
(1/2)ln(x^2) + 2ln(2) - (1/2)ln((x - 4)x)
The expression (1/2)ln(x^2) + 2ln(2) - (1/2)ln((x - 4)x) is the simplified single logarithm expression.