https://file.aiquickdraw.com/mathbot/file/1723471997467eeiw6p94.jpg

The equation in the image is:

\[
\log a + \log \left(\frac{a}{2}\right) = 2 \log b
\]

### Step 1: Use the logarithm properties
We can use the properties of logarithms to simplify the equation. The key properties we'll use are:
- \(\log(xy) = \log x + \log y\)
- \(\log\left(\frac{x}{y}\right) = \log x - \log y\)
- \(k \log x = \log x^k\)

Let's simplify the left-hand side (LHS) first:

\[
\log a + \log \left(\frac{a}{2}\right) = \log a + \left(\log a - \log 2\right)
\]
\[
= \log a + \log a - \log 2 = 2\log a - \log 2
\]

Now, the equation is:

\[
2\log a - \log 2 = 2\log b
\]

### Step 2: Combine the logarithms
Next, we'll isolate the logarithmic terms and combine them:

\[
2\log a - 2\log b = \log 2
\]

Factor out the common factor on the left side:

\[
2(\log a - \log b) = \log 2
\]

### Step 3: Apply logarithm subtraction property
\(\log a - \log b\) can be rewritten using the quotient rule of logarithms:

\[
2\log\left(\frac{a}{b}\right) = \log 2
\]

### Step 4: Simplify the equation
Divide both sides by 2:

\[
\log\left(\frac{a}{b}\right) = \frac{\log 2}{2}
\]

The logarithm on the right side can be rewritten as:

\[
\log\left(\frac{a}{b}\right) = \log\left(\sqrt{2}\right)
\]

### Step 5: Equate the arguments
Since the logarithms are equal, their arguments must be equal:

\[
\frac{a}{b} = \sqrt{2}
\]

### Step 6: Solve for \(a\)
Finally, solve for \(a\):

\[
a = b\sqrt{2}
\]

So, the solution to the equation is:

\[
a = b\sqrt{2}
\]

Would you like any additional details or have any questions?

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Here are some related questions you might consider:

1. What are the basic properties of logarithms?
2. How does the logarithm product rule apply in simplifying expressions?
3. Can you derive the logarithm quotient rule from the logarithm product rule?
4. What is the geometric interpretation of the logarithmic equation?
5. How do you solve logarithmic equations with different bases?
6. What are some real-life applications of logarithmic equations?
7. How do you express logarithms in terms of exponents?
8. What is the relationship between logarithms and exponential functions?

**Tip:** When solving logarithmic equations, always ensure the arguments of the logarithms are positive, as logarithms of non-positive numbers are undefined.

Learn how to solve the logarithmic equation Log(a) + Log(a/2) = 2 Log(b) with a detailed step-by-step solution. This problem covers key concepts in logarithms and logarithmic equations, including logarithm properties and equations. Suitable for advanced high school students.

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