Question

use properties of logarithms to expand each logarithmic expression as much as possible. evaluate logarithmic expressions using a calculator if possible. \\(\log_2(2x)\\) \\(\log_2(2x)=\square\\)

Explanation:

Step1: Recall the product rule of logarithms

The product rule of logarithms states that $\log_b(MN) = \log_b(M) + \log_b(N)$ for any positive real numbers $M$, $N$, and $b>0$, $b
eq1$.
For the expression $\log_2(2x)$, we can apply the product rule where $M = 2$ and $N=x$ and $b = 2$.
So, $\log_2(2x)=\log_2(2)+\log_2(x)$

Step2: Evaluate $\log_2(2)$

We know that for any base $b$, $\log_b(b) = 1$ because $b^1=b$. Here, $b = 2$, so $\log_2(2)=1$.
Substituting this value back into the expression from Step 1, we get $\log_2(2x)=1+\log_2(x)$

Answer:

$1 + \log_{2}(x)$