Sure, let's simplify the given logarithmic expression step by step.
We are given the expression:
[tex]\[ \ln [15 \ln (x-1) + \ln x - 5 \ln x] \][/tex]
### Step 1: Combine like terms inside the logarithm
First, let's combine the terms inside the logarithm to see if we can simplify it.
[tex]\[ 15 \ln (x-1) + \ln x - 5 \ln x \][/tex]
### Step 2: Simplify the coefficients of [tex]\(\ln x\)[/tex]
Notice that [tex]\(\ln x\)[/tex] appears twice inside the expression. Let's combine these terms:
[tex]\[ 15 \ln (x-1) + \ln x - 5 \ln x = 15 \ln (x-1) + (1 \ln x - 5 \ln x) \][/tex]
[tex]\[ = 15 \ln (x-1) - 4 \ln x \][/tex]
### Step 3: Substitute the simplified expression back into the logarithm
Now that we have simplified the terms inside the logarithm, we can rewrite the original expression:
[tex]\[ \ln [15 \ln (x-1) + \ln x - 5 \ln x] \][/tex]
[tex]\[ = \ln [15 \ln (x-1) - 4 \ln x] \][/tex]
Therefore, the simplified expression is:
[tex]\[ \boxed{\ln [15 \ln (x-1) - 4 \ln x]} \][/tex]
