$\log_b M + \log_b N = \log_b (M \times N)$
Let’s learn how to add logarithms. Don’t worry – it’s simple and straightforward.
What is a Logarithm?
A logarithm answers the question: To what exponent must we raise the base to get a certain number?
For example:
$\log_2 8 = 3$ because $2^3 = 8$.
How to Add Logarithms
When adding two logarithms with the same base, you can combine them into one logarithm by multiplying the numbers inside.
The Rule
\[\begin{align*} \log_b M + \log_b N = \log_b (M \times N) \end{align*}\]
Step-by-Step Example
Example: Simplify $\log_2 4 + \log_2 8$
Step 1:Â Multiply the numbers inside the logs:
\[\begin{align*} 4 \times 8 = 32 \end{align*}\]
Step 2:Â Combine into one logarithm:
\[\begin{align*} \log_2 4 + \log_2 8 = \log_2 32 \end{align*}\]
Step 3:Â Simplify if possible:
Since $2^5 = 32$, we have:
\[\begin{align*} \log_2 32 = 5 \end{align*}\]
Important Points to Remember
– The bases must be the same.
– Only multiply the numbers inside the logs.
– You can simplify further if possible.
Another Example
Example:Â Simplify $\log_{10} 5 + \log_{10} 2$
Solution:
1. Multiply the numbers inside:
\[\begin{align*} 5 \times 2 = 10 \end{align*}\]
2. Combine into one log:
\[\begin{align*} \log_{10} 5 + \log_{10} 2 = \log_{10} 10 \end{align*}\]
3. Simplify:
Since $\log_{10} 10 = 1$, the answer is $1$.
Practice Problem
Simplify $\log_3 3 + \log_3 9$
Solution:
1. Multiply inside numbers:
\[\begin{align*} 3 \times 9 = 27 \end{align*}\]
2. Combine:
\[\begin{align*} \log_3 3 + \log_3 9 = \log_3 27 \end{align*}\]
3. Simplify:
Since $3^3 = 27$, we have $\log_3 27 = 3$
Conclusion
Adding logs is easy when you remember to multiply the numbers inside.
\[ \begin{array}{|c|} \hline \text{\Large Adding Logs: } \log_b M + \log_b N = \log_b (M \times N) \\ \hline \end{array} \]