Use the properties of logarithms to rewrite as the difference of two logs: log_(5)45 -log_(5)2 log_(5)50-log_(5)5 (log45)/(log5) log_(5)15-

## Step 1The problem asks us to rewrite the given logarithmic expression using the properties of logarithms. The properties of logarithms that we will use are:### **The Product Rule**: \( \log_b(MN) = \log_b(M) + \log_b(N) \)### **The Quotient Rule**: \( \log_b(M/N) = \log_b(M) - \log_b(N) \)### **The Change of Base Formula**: \( \log_b(M) = \frac{\log_c(M)}{\log_c(b)} \)## Step 2We are given the expression . We can break down 45 into 9 and 5, which are both powers of 5. So, we can rewrite this as:### \( \log_{5}45 = \log_{5}(9*5) \)## Step 3Using the Product Rule, we can split this into two separate logarithms:### \( \log_{5}(9*5) = \log_{5}9 + \log_{5}5 \)## Step 4Next, we have the expression . This can be rewritten as:### \( -\log_{5}2 = \log_{5}(1/2) \)## Step 5We can then use the Quotient Rule to split this into two separate logarithms:### \( \log_{5}(1/2) = \log_{5}1 - \log_{5}2 \)## Step 6Finally, we have the expression . This can be rewritten as:### \( \log_{5}50 - \log_{5}5 = \log_{5}(50/5) \)## Step 7Using the Quotient Rule, we can split this into two separate logarithms:### \( \log_{5}(50/5) = \log_{5}10 - \log_{5}5 \)