$\log_b M + \log_b N = \log_b (M \times N)$

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: Multiply the numbers inside the logs:

\[\begin{align*} 4 \times 8 = 32 \end{align*}\]

Step 2: Combine into one logarithm:

\[\begin{align*} \log_2 4 + \log_2 8 = \log_2 32 \end{align*}\]

Step 3: 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: 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.

\[ \begin{array}{|c|} \hline \text{\Large Adding Logs: } \log_b M + \log_b N = \log_b (M \times N) \\ \hline \end{array} \]