# Thread: Using properties of logarithms to compute

1. ## Using properties of logarithms to compute

Given ln 2 = 0.6931 and ln 3 = 1.0986, use properties of logarithms to compute:

ln square root(8/27), Sorry I have no idea how to put a square root sign in.

Thank you very much

2. ## Re: Using properties of logarithms to compute

Can you write the argument for the log function as a fraction involving only 2 and 3 raised to some power?

3. ## Re: Using properties of logarithms to compute

 $\displaystyle \ln{\sqrt{\frac{8}{27}}}$ Given $\displaystyle = \ln{\left\frac{8}{27}\right} ^\frac{1}{2}$ Equivalent fractional exponent $\displaystyle =\frac{\ln{\frac{8}{27}}}{2}$ Log property: $\displaystyle \log_b{a^n} = n\log_b{a}$ $\displaystyle =\frac{\ln{8} - \ln{27}}{2}$ Log property: $\displaystyle \log_b{\frac{a}{n}} = \log_b{a} - \log_b{n}$ $\displaystyle =\frac{\ln{2^3} - \ln{3^3}}{2}$ $\displaystyle 8 = 2^3$ and $\displaystyle 27 = 3^3$ $\displaystyle =\frac{3\ln{2} - 3\ln{3}}{2}$ Log property: $\displaystyle \log_b{a^n} = n\log_b{a}$ $\displaystyle =\frac{3(0.6931) - 3(1.0986)}{2}$ Substitute $\displaystyle \ln 2$ and $\displaystyle \ln 3$ $\displaystyle \approx -0.60825$ Simplify with calculator