Questions: Expand the expression. If possible, write your answer without exponents. ln (s^7 / t^5 sqrt(u^3)) ln (s^7 / t^5 sqrt(u^3)) =

Solution Steps

We start with the expression: \[ \ln \frac{s^{7}}{t^{5} \sqrt{u^{3}}} \] Using the logarithm property \(\ln \frac{a}{b} = \ln a - \ln b\), we can rewrite the expression as: \[ \ln s^{7} - \ln \left(t^{5} \sqrt{u^{3}}\right) \]

Next, we expand \(\ln \left(t^{5} \sqrt{u^{3}}\right)\) using the logarithm property \(\ln(ab) = \ln a + \ln b\): \[ \ln \left(t^{5} \sqrt{u^{3}}\right) = \ln t^{5} + \ln \sqrt{u^{3}} \]

Now, we simplify the exponents using the logarithm property \(\ln a^{b} = b \ln a\): \[ \ln s^{7} = 7 \ln s \] \[ \ln t^{5} = 5 \ln t \] \[ \ln \sqrt{u^{3}} = \ln u^{3/2} = \frac{3}{2} \ln u \]

Substituting the simplified expressions back into the equation, we get: \[ 7 \ln s - \left(5 \ln t + \frac{3}{2} \ln u\right) \] Distribute the negative sign: \[ 7 \ln s - 5 \ln t - \frac{3}{2} \ln u \]

Final Answer