Write the expression as a single log. Express powers as factors
ln(x/x-5) + ln(x+5/x) - ln(x^2-25)
I am sure you meant to type
ln( x/(x-5) ) + ln( (x+5)/x ) - ln(x^2 - 25)
= lnx - ln(x-5) + ln(x+5) - lnx - (ln(x+5) + ln(x-5)
= -ln(x-5) + ln(x+5) - ln(x+5) - ln(x-5)
= -2ln(x-5)
To express the given expression as a single logarithm, we can use the properties of logarithms. The first step is to simplify each individual logarithmic term using those properties.
Step 1: Simplify each logarithmic term separately.
- ln(x/x-5) can be rewritten as ln(x) - ln(x-5).
- ln(x+5/x) can be rewritten as ln(x+5) - ln(x).
- ln(x^2-25) can be rewritten as ln((x-5)(x+5)).
Step 2: Combine the logarithmic terms using addition and subtraction rules of logarithms.
Now, let's combine these simplified logarithmic terms.
The expression becomes ln(x) - ln(x-5) + ln(x+5) - ln((x-5)(x+5)).
Step 3: Simplify further using the quotient rule.
Applying the quotient rule of logarithms, we can rewrite the expression as a single logarithm.
Using the quotient rule, ln(a) - ln(b) = ln(a/b), the expression becomes:
ln((x * (x+5))/((x-5) * (x+5))).
Note that (x+5) cancels out in the numerator and denominator, so we get:
ln(x/(x-5)).
Therefore, the given expression, ln(x/x-5) + ln(x+5/x) - ln(x^2-25), can be simplified and written as ln(x/(x-5)).