Questions: Use properties of logarithms to expand the logarithmic expression as much as possible. Assume s and t are positive real numbers. log(sqrt[3](s/t))

Use properties of logarithms to expand the logarithmic expression as much as possible. Assume s and t are positive real numbers.
log(sqrt[3](s/t))

Transcript text: Use properties of logarithms to expand the logarithmic expression as much as possible. Assume s and $t$ are positive re numbers. \[ \log \sqrt[3]{\frac{s}{t}} \] (Type an exact answer in simplified form.)

Solution

Solution Steps

To expand the logarithmic expression $\log \sqrt[3]{\frac{s}{t}}$, we can use the properties of logarithms. First, recognize that the cube root can be expressed as a fractional exponent. Then, apply the power rule of logarithms to bring the exponent in front of the log. Next, use the quotient rule to separate the log of a fraction into the difference of two logs.

Step 1: Apply the Logarithm of a Root Property

The given expression is:

\[ \log \sqrt[3]{\frac{s}{t}} \]

We can use the property of logarithms that states:

\[ \log \sqrt[n]{x} = \frac{1}{n} \log x \]

In this case, $ n = 3 $ and $ x = \frac{s}{t} $. Therefore, we have:

\[ \log \sqrt[3]{\frac{s}{t}} = \frac{1}{3} \log \left( \frac{s}{t} \right) \]

Step 2: Apply the Quotient Rule of Logarithms

The quotient rule of logarithms states:

\[ \log \left( \frac{a}{b} \right) = \log a - \log b \]

Applying this to our expression:

\[ \frac{1}{3} \log \left( \frac{s}{t} \right) = \frac{1}{3} (\log s - \log t) \]