6. Estimate the value to three decimal places. a) log_(3)53 c) log_(6)159 b) log_(4)(1)/(10) d) log_(15)1456

## Step 1: Change of Base Formula<br />### We will use the change of base formula, which states $\log_b{a} = \frac{\log_c{a}}{\log_c{b}}$, where $c$ can be any base. We will use base 10 (common logarithm) for our calculations.<br /><br />## Step 2: Calculate $log_{3}53$<br />### Using the change of base formula: $log_{3}53 = \frac{log_{10}53}{log_{10}3} \approx \frac{1.72427587}{0.47712125} \approx 3.613$<br /><br />## Step 3: Calculate $log_{4}\frac{1}{10}$<br />### Using the change of base formula: $log_{4}\frac{1}{10} = \frac{log_{10}\frac{1}{10}}{log_{10}4} = \frac{log_{10}1 - log_{10}10}{log_{10}4} = \frac{0 - 1}{0.60205999} \approx -1.661$<br /><br />## Step 4: Calculate $log_{6}159$<br />### Using the change of base formula: $log_{6}159 = \frac{log_{10}159}{log_{10}6} \approx \frac{2.20139712}{0.77815125} \approx 2.829$<br /><br />## Step 5: Calculate $log_{15}1456$<br />### Using the change of base formula: $log_{15}1456 = \frac{log_{10}1456}{log_{10}15} \approx \frac{3.16286109}{1.17609126} \approx 2.689$