x=3×log 
2
​
 ( 
25
36
​
 )+3×log 
2
​
 ( 
27
6
​
 )+2×log 
0,5
​
 ( 
25
16
​
 )

Let's break down the given equation step by step:

\[
x = 3 \times \log_2\left(\frac{25}{36}\right) + 3 \times \log_2\left(\frac{27}{6}\right) + 2 \times \log_{0.5}\left(\frac{25}{16}\right)
\]

### Step 1: Use the properties of logarithms
We can use the power rule of logarithms to simplify the coefficients in front of the logarithms:

\[
3 \times \log_2\left(\frac{25}{36}\right) = \log_2\left(\frac{25}{36}\right)^3 = \log_2\left(\frac{25^3}{36^3}\right)
\]
\[
3 \times \log_2\left(\frac{27}{6}\right) = \log_2\left(\frac{27}{6}\right)^3 = \log_2\left(\frac{27^3}{6^3}\right)
\]
\[
2 \times \log_{0.5}\left(\frac{25}{16}\right) = \log_{0.5}\left(\frac{25^2}{16^2}\right)
\]

### Step 2: Simplify the expressions
Now, let's compute the powers:
- \( 25^3 = 15625 \)
- \( 36^3 = 46656 \)
- \( 27^3 = 19683 \)
- \( 6^3 = 216 \)
- \( 25^2 = 625 \)
- \( 16^2 = 256 \)

Thus, the equation becomes:

\[
x = \log_2\left(\frac{15625}{46656}\right) + \log_2\left(\frac{19683}{216}\right) + \log_{0.5}\left(\frac{625}{256}\right)
\]

### Step 3: Use the sum of logarithms
Using the logarithm property that the sum of logarithms is the logarithm of the product, we can combine the first two terms:

\[
x = \log_2\left(\frac{15625}{46656} \times \frac{19683}{216}\right) + \log_{0.5}\left(\frac{625}{256}\right)
\]

We can now compute these expressions numerically.

Let me calculate this value for you.The value of \( x \) is approximately \( 3.64 \).

Would you like more details or have any questions?

Here are 5 related questions to consider:
1. What are the logarithmic properties used in this problem?
2. How would the equation change if the base of the logarithm was different?
3. Can you solve similar equations with natural logarithms (\( \ln \))?
4. What is the effect of changing the base from \( \log_2 \) to \( \log_{0.5} \)?
5. How do logarithmic scales apply to real-world problems like sound or earthquakes?

**Tip**: When solving logarithmic equations, remember that using properties like the power rule or the product rule can simplify your calculations significantly.

x = 3 × log2(25/36) + 3 × log2(27/6) + 2 × log0.5(25/16)

Learn how to solve the logarithmic equation x = 3 × log2(25/36) + 3 × log2(27/6) + 2 × log0.5(25/16). This problem demonstrates the application of logarithmic properties such as the power rule and product rule. Suitable for high school students (Grades 10-12).

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