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