Write as a Single Logarithm ( log of s)/( log of r)+8 log base r of s-3 log base r of t

$\frac{\log (s)}{\log (r)} + 8 \log_{r} (s) - 3 \log_{r} (t)$

Step 1

Simplify each term.

Simplify $8 \log_{r} (s)$ by moving $8$ inside the logarithm.

$\frac{\log (s)}{\log (r)} + \log_{r} (s^{8}) - 3 \log_{r} (t)$

Simplify $- 3 \log_{r} (t)$ by moving $3$ inside the logarithm.

$\frac{\log (s)}{\log (r)} + \log_{r} (s^{8}) - \log_{r} (t^{3})$

Step 2

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\frac{\log (s)}{\log (r)} + \log_{r} (\frac{s^{8}}{t^{3}})$

Step 3

To write $\log_{r} (\frac{s^{8}}{t^{3}})$ as a fraction with a common denominator, multiply by $\frac{\log (r)}{\log (r)}$ .

$\frac{\log (s)}{\log (r)} + \frac{\log_{r} (\frac{s^{8}}{t^{3}}) \log (r)}{\log (r)}$

Step 4

Combine the numerators over the common denominator.

$\frac{\log (s) + \log_{r} (\frac{s^{8}}{t^{3}}) \log (r)}{\log (r)}$