Question

Suppose that ln 2 = a and ln 3 = b. Use properties of logarithms to write each logarithm in terms of a and b.

$\ln \sqrt[5]{6}$

Short answer

The result is$\frac{1}{5}\left(a+b\right)$.

Step-by-step solution

Step 1. Given information.

Consider the given information.

$\ln \sqrt[5]{6}$

Step 2. Calculation. 

Apply the properties of logarithms.

$\begin{array}{l}\ln \sqrt[5]{6}=\ln 6^{\frac{1}{5}}\\ =\frac{1}{5}\ln \left(2·3\right)\\ =\frac{1}{5}\left[\ln 2+\ln 3\right]\end{array}$

Step 3. Substitute the given values.

Substitute ln 2 = a and ln 3 = bin the obtained result.

$\frac{1}{5}\left[\ln 2+\ln 3\right]=\frac{1}{5}\left(a+b\right)$