Question

Log12 9 + 2 log 12 4

Answer 1

$=log_{12}9+log_{12}4^{2}=log_{12}3^{2}+log_{12}4^{2}=log_{12}(3^{2}*4^{2})=log_{12}(3*4)^{2}=2*log_{12}(3*4)=2*log_{12}12=2*1=2$

Answer 2

To simplify the expression log₁₂(9) + 2log₁₂(4), we'll use some logarithmic properties:

1. log(a) + log(b) = log(ab)
2. n * log(a) = log(aⁿ)

Let's proceed step-by-step:

1. log₁₂(9) can be simplified using the fact that 9 is 3²: log₁₂(9) = log₁₂(3²)

2. Apply property 2 to simplify log₁₂(3²): log₁₂(9) = 2 * log₁₂(3)

3. log₁₂(4) can be simplified using the fact that 4 is 2²: log₁₂(4) = log₁₂(2²)

4. Apply property 2 to simplify log₁₂(2²): log₁₂(4) = 2 * log₁₂(2)

Now, the expression becomes:

2 * log₁₂(3) + 2 * log₁₂(2)

Next, we can factor out the common factor of 2:

2 * (log₁₂(3) + log₁₂(2))

Now, we can apply property 1 to combine the two logarithms inside the parentheses:

2 * log₁₂(3 * 2)

Simplify the expression inside the logarithm:

2 * log₁₂(6)

And that's the final simplified expression:

2 * log₁₂(6)