https://file.aiquickdraw.com/mathbot/file/1726064541413bafkkvci.jpg

Let's focus on solving the three circled math problems (numbered 17, 18, and 19) in the image. I will start by analyzing each one and providing solutions.

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**Problem 17:**

\[
\text{If } \log a = 1 \text{ and } \log b = 2, \text{ find the value of } \log \frac{a}{b^2}.
\]

We can use the properties of logarithms:

\[
\log \frac{a}{b^2} = \log a - \log b^2 = \log a - 2\log b
\]

Substitute the given values:

\[
\log \frac{a}{b^2} = 1 - 2(2) = 1 - 4 = -3
\]

So, the answer is **-3** (Option B).

---

**Problem 18:**

\[
\text{A circle has a radius of } \log d^2 \text{ and the circumference is } \log b, \text{ find } \log b.
\]

We know that the circumference of a circle is given by:

\[
\text{Circumference} = 2\pi \times \text{radius}
\]

Substitute the radius:

\[
\log b = 2\pi \times \log d^2
\]

We can simplify using logarithmic properties:

\[
\log d^2 = 2 \log d
\]

Thus:

\[
\log b = 2\pi \times 2 \log d = 4\pi \log d
\]

So, the answer is **4π** (Option C).

---

**Problem 19:**

\[
\text{If } \log \left( \frac{a}{b} \right) = 4, \text{ find the value of } \log \left( \sqrt{a} + \sqrt{b} \right).
\]

We use the logarithmic property:

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

So, given \(\log \left( \frac{a}{b} \right) = 4\), we assume that \(\log a - \log b = 4\). Now, for \(\log \left( \sqrt{a} + \sqrt{b} \right)\), we rewrite:

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

Using the property \(\log \sqrt{x} = \frac{1}{2} \log x\), we simplify:

\[
\log \left( \sqrt{a} + \sqrt{b} \right) = \frac{1}{2} \log a + \frac{1}{2} \log b
\]

Thus, the value is:

\[
\frac{1}{2}(\log a + \log b) = \frac{1}{2} \times 4 = 2
\]

So, the answer is **2** (Option B).

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Would you like any further explanation on these solutions?

Here are 5 related questions for practice:
1. What is the value of \(\log \frac{b^3}{a}\) if \(\log a = 1\) and \(\log b = 2\)?
2. If the area of a circle is \(\log r^2\), what is the relationship between the radius and the area in logarithmic terms?
3. If \(\log \frac{a}{b^4} = 3\), how would you express \(\log a\) in terms of \(\log b\)?
4. Solve \(\log (x^2) + \log (y^2) = 5\) in terms of \(\log x + \log y\).
5. Find the value of \(\log \frac{a^2}{b}\) if \(\log a = 3\) and \(\log b = 1\).

**Tip:** When working with logarithms, remember that simplifying expressions using basic properties like the product, quotient, and power rules often makes the problem more manageable.

Solve the following logarithmic problems: Problem 17, Problem 18, and Problem 19.

This page provides detailed solutions to three logarithmic problems involving log properties, including solving for log(a/b^2), finding log b in terms of a circle's radius, and solving log expressions with square roots. Suitable for students in Grades 10-12.

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