How do you expand log(x(x^3+9)^(-1/2))?

Mar 20, 2016

We must use the log properties to simplify.

Explanation:

1) Use ${\log}_{n} \left(a \times b\right) = {\log}_{n} a + {\log}_{n} b$ to seperate the expression into two logarithmic expressions.

= $\log x + \log {\left({x}^{3} + 9\right)}^{- \frac{1}{2}}$

2). Use $\log {a}^{x} = x \log a$ to get rid of the exponent.

Since the exponent is negative, we should first use the exponential property ${a}^{- n} = \frac{1}{{a}^{n}}$

=$\log x + \log \left(\frac{1}{{x}^{3} + 9} ^ \left(\frac{1}{2}\right)\right)$

=$\log x + \frac{1}{2} \log \left(\frac{1}{{x}^{3} + 9}\right)$

3). We now use the rule ${\log}_{a} \left(\frac{n}{m}\right) = {\log}_{a} n - {\log}_{a} m$

=$\log x + \frac{1}{2} \left(\log 1 - \log \left({x}^{3} + 9\right)\right)$

=$\log x + \frac{1}{2} \log 1 - \frac{1}{2} \log \left({x}^{3} + 9\right)$

Since $\log 1 = 0$, we are left with $\log x - \frac{1}{2} \log \left({x}^{3} + 9\right)$

We could have just used the negative $- \frac{1}{2}$ at step 2, but I took the opportunity to show you an additional log rule.

Hopefully this helps!