# Simplify 5lnx +21lnx^3-3lnx^3+lnx^(1/2)?

Mar 3, 2017

$5 \ln x + 21 \ln {x}^{3} - 3 \ln {x}^{3} + \ln {x}^{\frac{1}{2}} = \frac{119}{2} \ln x$

#### Explanation:

We can use the identities $\ln a + \ln b - \ln c = \ln \left(\frac{a b}{c}\right)$ and $\ln {a}^{b} = b \ln a$.

Hence,

$5 \ln x + 21 \ln {x}^{3} - 3 \ln {x}^{3} + \ln {x}^{\frac{1}{2}}$

= $5 \ln x + 21 \times 3 \ln x - 3 \times 3 \ln x + \frac{1}{2} \ln x$

= $5 \ln x + 63 \ln x - 9 \ln x + \frac{1}{2} \ln x$

= $\ln x \left(5 + 63 - 9 + \frac{1}{2}\right)$

= $\ln x \left(59 + \frac{1}{2}\right)$

= $\frac{119}{2} \ln x$