Question

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

Answer 1

We must use the log properties to simplify.

Explanation:

1) Use `log_n(a xx b) = log_na + log_nb` to seperate the expression into two logarithmic expressions.

= `logx + log(x^3 + 9)^(-1/2)`

2). Use `loga^x = xloga` to get rid of the exponent.

Since the exponent is negative, we should first use the exponential property `a^(-n) = 1/(a^n)`

=`logx + log(1/(x^3 + 9)^(1/2))`

=`logx + 1/2log(1/(x^3 + 9))`

3). We now use the rule `log_a(n/m) = log_an - log_am`

=`logx + 1/2(log1 - log(x^3 + 9))`

=`logx + 1/2log1 - 1/2log(x^3 + 9)`

Since `log1 = 0`, we are left with `logx - 1/2log(x^3 + 9)`

We could have just used the negative `-1/2` at step 2, but I took the opportunity to show you an additional log rule.

Hopefully this helps!