# How do you condense -5 log_b (y) - 6 log_b (z) + 1/2 log_b (x+4)?

Jul 16, 2016

=${\log}_{b} \left[\frac{\sqrt{\left(x + 4\right)}}{{y}^{11}}\right]$

#### Explanation:

All the terms are logs to the same base.

Use the power law first:

$- 5 {\log}_{b} \left(y\right) - 6 {\log}_{b} \left(z\right) + \frac{1}{2} {\log}_{b} \left(x + 4\right)$

=$- {\log}_{b} {\left(y\right)}^{5} - {\log}_{b} {\left(z\right)}^{6} + {\log}_{b} {\left(x + 4\right)}^{\frac{1}{2}}$

If logs are added, the numbers are multiplied. if logs are subtracted, the numbers are divided.

=${\log}_{b} \left[\frac{{\left(x + 4\right)}^{\frac{1}{2}}}{{y}^{5} \times {y}^{6}}\right]$

=${\log}_{b} \left[\frac{\sqrt{\left(x + 4\right)}}{{y}^{11}}\right]$