https://file.aiquickdraw.com/mathbot/file/1726153025665v4bka18f.jpg

Let's solve the questions shown in the image.

### (a) Given \( 2 \log_2 y = \log_2 (x + 4) \), express \( y \) in terms of \( x \).

To solve for \( y \):

1. Start with the equation:
   \[
   2 \log_2 y = \log_2 (x + 4)
   \]

2. Divide both sides by 2 to isolate \( \log_2 y \):
   \[
   \log_2 y = \frac{1}{2} \log_2 (x + 4)
   \]

3. Use the property of logarithms \( \frac{1}{2} \log_b A = \log_b (A^{1/2}) \) to simplify:
   \[
   \log_2 y = \log_2 \sqrt{x + 4}
   \]

4. Since the logarithms have the same base, you can conclude:
   \[
   y = \sqrt{x + 4}
   \]

Thus, the expression for \( y \) in terms of \( x \) is:
\[
y = \sqrt{x + 4}
\]

---

### (b) Solve the equation \( 4^x = 80 \cdot 4^{1-x} \).

To solve for \( x \):

1. Start with the equation:
   \[
   4^x = 80 \cdot 4^{1 - x}
   \]

2. Express \( 4^{1 - x} \) as \( 4 \cdot 4^{-x} \) using the laws of exponents:
   \[
   4^x = 80 \cdot 4 \cdot 4^{-x}
   \]

3. Simplify the right-hand side:
   \[
   4^x = 320 \cdot 4^{-x}
   \]

4. Multiply both sides by \( 4^x \) to eliminate the negative exponent:
   \[
   (4^x)^2 = 320
   \]

5. Simplify:
   \[
   16^x = 320
   \]

6. Take the logarithm of both sides (log base 2 is convenient):
   \[
   \log_2 (16^x) = \log_2 320
   \]

7. Use the property \( \log_b (A^n) = n \log_b A \):
   \[
   x \cdot \log_2 16 = \log_2 320
   \]

8. Since \( \log_2 16 = 4 \), the equation becomes:
   \[
   4x = \log_2 320
   \]

9. Now, evaluate \( \log_2 320 \). We know \( 320 = 2^8 \times 5 \), so:
   \[
   \log_2 320 = 8 + \log_2 5
   \]
   Using approximate values for \( \log_2 5 \) (about 2.32), we get:
   \[
   \log_2 320 \approx 8 + 2.32 = 10.32
   \]

10. Solve for \( x \):
   \[
   x = \frac{10.32}{4} \approx 2.58
   \]

Thus, the approximate solution is:
\[
x \approx 2.58
\]

---

Let me know if you'd like further clarification on any step or have other questions. Here are some related questions for practice:

1. Can you generalize the approach for any equation of the form \( a^x = b \cdot a^{c-x} \)?
2. What is the relationship between logarithms and exponents, and how can it help in solving such equations?
3. How would you solve \( 4^x = 160 \cdot 4^{-x} \)?
4. If \( 2 \log_2 y = \log_2 (x + 9) \), how would you express \( y \) in terms of \( x \)?
5. What are the laws of logarithms, and how can they be applied to equations involving logarithmic expressions?

**Tip:** When working with logarithmic equations, always remember to use the properties of logarithms to simplify expressions before solving them.

Given 2log2y = log2(x + 4), express y in terms of x. Solve the equation 4^x = 80 ⋅ 4^(1-x).

Learn how to express y in terms of x using logarithmic properties and solve an exponential equation 4^x = 80 ⋅ 4^(1-x). Detailed step-by-step solutions and application of logarithmic and exponential formulas. Ideal for high school students (Grades 9-12).

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation