8. Use the laws of logarithms to evaluate. a) log_(6)42-log_(6)7 b) log_(3)5+log_(3)18-log_(3)10 c) log_(7)sqrt [3](49) d) 2log_(4)8

## Step 1: Quotient Rule<br />### The difference of logarithms with the same base is equivalent to the logarithm of the quotient. $log_{b}m - log_{b}n = log_{b}(\frac{m}{n})$<br />## Step 2: Simplify the expression<br />### Substitute the given values into the quotient rule: $log_{6}42 - log_{6}7 = log_{6}(\frac{42}{7}) = log_{6}6 = 1$<br />## Step 3: Product and Quotient Rule<br />### The sum of logarithms with the same base is equivalent to the logarithm of the product, and the difference is the logarithm of the quotient. $log_{b}m + log_{b}n = log_{b}(mn)$ and $log_{b}m - log_{b}n = log_{b}(\frac{m}{n})$<br />## Step 4: Simplify the expression<br />### Substitute the given values into the product and quotient rule: $log_{3}5 + log_{3}18 - log_{3}10 = log_{3}(\frac{5 \cdot 18}{10}) = log_{3}9 = 2$<br />## Step 5: Power Rule<br />### The logarithm of a number raised to a power is equal to the power multiplied by the logarithm of the number. $log_{b}m^n = nlog_{b}m$<br />## Step 6: Simplify the expression<br />### Rewrite the expression using the power rule and simplify: $log_{7}\sqrt[3]{49} = log_{7}49^{\frac{1}{3}} = \frac{1}{3}log_{7}49 = \frac{1}{3}log_{7}7^2 = \frac{1}{3} \cdot 2 = \frac{2}{3}$<br />## Step 7: Power Rule and Simplify<br />### Apply the power rule and simplify the expression: $2log_{4}8 = log_{4}8^2 = log_{4}64 = log_{4}4^3 = 3$