Log√2 12 + Log√2 3= Log√2 (12×3)= Log√2 36=?

To solve the given logarithmic expression:

\[
\log_{\sqrt{2}}{12} + \log_{\sqrt{2}}{3} = \log_{\sqrt{2}}{(12 \times 3)} = \log_{\sqrt{2}}{36}
\]

### Step 1: Use the logarithmic product rule:
The product rule of logarithms states:

\[
\log_b{a} + \log_b{c} = \log_b{(a \times c)}
\]

Thus:

\[
\log_{\sqrt{2}}{12} + \log_{\sqrt{2}}{3} = \log_{\sqrt{2}}{(12 \times 3)} = \log_{\sqrt{2}}{36}
\]

### Step 2: Rewrite the logarithmic base:
Since \(\sqrt{2} = 2^{1/2}\), the logarithm can be rewritten as:

\[
\log_{\sqrt{2}}{36} = \frac{\log_2{36}}{\log_2{\sqrt{2}}}
\]

We know that:

\[
\log_2{\sqrt{2}} = \log_2{(2^{1/2})} = \frac{1}{2}
\]

Thus:

\[
\log_{\sqrt{2}}{36} = \frac{\log_2{36}}{1/2} = 2 \log_2{36}
\]

### Step 3: Simplify \(\log_2{36}\):
We can approximate \(\log_2{36}\) by recognizing that:

\[
36 \approx 32 \quad \text{and} \quad \log_2{32} = 5
\]

So:

\[
\log_2{36} \approx 5.17
\]

### Step 4: Calculate the final value:
Now we can calculate the approximate value:

\[
2 \log_2{36} \approx 2 \times 5.17 = 10.34
\]

Thus, \(\log_{\sqrt{2}}{36} \approx 10.34\).

Would you like more details or have any questions?

Here are some related questions to consider:
1. How do you convert logarithms to different bases?
2. What is the change of base formula in logarithms?
3. How do you approximate logarithms for non-exact powers?
4. What are the properties of logarithms involving roots?
5. How can logarithms be applied to exponential equations?

**Tip:** When working with logarithms involving roots, it's often helpful to express the root as a fractional exponent to simplify calculations.

Log√2 12 + Log√2 3 = Log√2 (12×3) = Log√2 36 =?

Solve the logarithmic expression Log√2 12 + Log√2 3 using the product rule of logarithms and change of base formula. Understand how to rewrite logarithmic bases and approximate logarithmic values. Suitable for Grades 10-12.

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