Let $\log_a x = b$. Then $x = a^b$.

Taking logarithm to base $c$ on both sides:

$\log_c x = \log_c a^b$

$\log_c x = b \log_c a$

$b = \frac{\log_c x}{\log_c a}$

$\log_a x = \frac{\log_c x}{\log_c a}$

If $c = b$:

$\log_a x = \frac{\log_b x}{\log_b a}$

Example 9.7: Evaluating logarithms

a) $\log_8 2$

b) $\log_7 3$

Solution:

a) $\log_8 2 = \dfrac{\log_2 2}{\log_2 8} = \dfrac{1}{3}$

b) $\log_7 3 = \dfrac{\log_{10} 3}{\log_{10} 7} \approx 0.564575$ or

$\log_7 3 = \dfrac{\ln 3}{\ln 7} \approx 0.564575$

1. $\log_b b = 1$
2. $\log_b a^x = x \log_b a$
3. $\log_b (xy) = \log_b x + \log_b y$
4. $\log_b (x/y) = \log_b x - \log_b y$
5. $\log_b 1 = 0$
6. $\log_a x = \dfrac{\log_c x}{\log_c a}$
7. $\log_a x = \dfrac{1}{\log_x a}$

1. $\ln 1 = 0$
2. $\ln a^x = x \ln a$
3. $\ln(xy) = \ln x + \ln y$
4. $\ln(x/y) = \ln x - \ln y$
5. $\ln e = 1$