https://file.aiquickdraw.com/mathbot/file/1725877727601kz73o093.jpg

Let's solve the equations from the image.

### 4. Solve for \(x\):

**(a)** \( 4^{2x-1} = 1024 \)

We can rewrite both sides using powers of 2:
\[
4^{2x-1} = (2^2)^{2x-1} = 2^{4x - 2}
\]
Also, \( 1024 = 2^{10} \).

Now, we have:
\[
2^{4x-2} = 2^{10}
\]

Equating the exponents:
\[
4x - 2 = 10
\]

Solve for \(x\):
\[
4x = 12 \quad \Rightarrow \quad x = 3
\]

**(b)** \( 3^{2(9x - 2)} = 27^{x+1} \)

Since \( 27 = 3^3 \), we can rewrite the equation as:
\[
3^{2(9x - 2)} = 3^{3(x+1)}
\]

Equating the exponents:
\[
2(9x - 2) = 3(x + 1)
\]

Expand both sides:
\[
18x - 4 = 3x + 3
\]

Solve for \(x\):
\[
18x - 3x = 3 + 4 \quad \Rightarrow \quad 15x = 7 \quad \Rightarrow \quad x = \frac{7}{15}
\]

**(c)** If \( 2^y = 24 \), determine the value of \(x\) and \(z\) if \(x = 2^{y - 2}\) and \(z = 3(2^{y + 1})\).

First, solve for \(y\):
\[
2^y = 24
\]
Take the natural logarithm of both sides:
\[
y \ln 2 = \ln 24 \quad \Rightarrow \quad y = \frac{\ln 24}{\ln 2}
\]
Using approximate logarithm values:
\[
y = \frac{\ln 24}{\ln 2} \approx \frac{3.1781}{0.6931} \approx 4.585
\]

Now, solve for \(x\):
\[
x = 2^{y - 2} = 2^{4.585 - 2} = 2^{2.585} \approx 6.059
\]

Finally, solve for \(z\):
\[
z = 3(2^{y+1}) = 3(2^{4.585 + 1}) = 3(2^{5.585}) \approx 3 \times 48.09 = 144.27
\]

---

Do you have any questions or would you like further details?

### Related Questions:
1. How can I simplify exponents when both sides have powers of the same base?
2. How do logarithmic properties help in solving exponential equations?
3. What is the general process for converting a non-power base to one with a common base?
4. Can we solve for other variables using natural logarithms for bases that aren't powers of 2 or 3?
5. How can I apply this method to solving more complex exponent problems?

**Tip**: When solving exponential equations, always try to express both sides with the same base for easier comparison of exponents.

Solve for x:
(a) 4^(2x-1) = 1024
(b) 3^(2(9x-2)) = 27^(x+1)
(c) If 2^y = 24, determine the value of x and z if x = 2^(y-2) and z = 3(2^(y+1))

Learn how to solve exponential equations such as 4^(2x-1) = 1024 and determine the values of x and z from complex power expressions. Explore logarithmic techniques and algebraic steps with detailed solutions. Suitable for high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation