Working with Logarithmic Functions

Video lesson · from 20:222 worked examples

One lesson video covers all of Exponentials & Logarithms, so it opens at 20:22 for this topic — not from the beginning.

If we have the sum of several terms involving logarithmic functions then we can combine the terms in one step.

Clear coefficients first: $2\log_a 3$ must become $\log_a 9$ before it can be combined with anything. Trying to add while a coefficient sits out in front is the usual error.
Then add and subtract the arguments: sums become products, differences become quotients.
Same base throughout: check before you start — mixed bases cannot be combined.
Simplify the final argument: leave $\log_a 12$ rather than $\log_a (4 \times 3)$ .

Example 1

Evaluate $\log_a 10 + \log_a 6 - \log_a 5$

\log_a 10 + \log_a 6 - \log_a 5 = \log_a\left(\frac{10 \times 6}{5}\right) = \log_a\left(\frac{60}{5}\right) = \log_a 12

Example 2

Evaluate $\log_a 4 + 2\log_a 3$

\log_a 4 + 2\log_a 3 = \log_a 4 + \log_a(3^2) = \log_a 4 + \log_a 9 = \log_a(4 \times 9) = \log_a 36