The difference of logarithms



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

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


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.

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