# The difference of logarithms

## Statement

The difference of two logarithms with the same base is just one logarithm with the inside parts being divided.

$$\log_a(b) - \log_a(c) = \log\left(\frac{b}{c}\right)$$

## Proof

Write the difference as a sum.

$$\log_a(b) + -1 * \log_a(c)$$

Use the logarithm power rule to write the following.

$$\log_a(b) + \log_a(c^{-1})$$

Now use the logarithm sum rule to combine the logarithms.

$$\log_a(b * c^{-1})$$

Finally, write the negative exponent as a fraction.

$$\log_a\left(\frac{b}{c}\right)$$