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The image contains four exponential equations to solve. Let's work through each one step by step, rounding the answers to three decimal places as requested.

### 1. \( 8^{x+3} = 22 \)

Taking the logarithm of both sides:

\[
(x+3) \log(8) = \log(22)
\]

Solve for \(x\):

\[
x+3 = \frac{\log(22)}{\log(8)}
\]

Then:

\[
x = \frac{\log(22)}{\log(8)} - 3
\]

### 2. \( 1.23^{x+4} = 35 \)

Taking the logarithm of both sides:

\[
(x+4) \log(1.23) = \log(35)
\]

Solve for \(x\):

\[
x+4 = \frac{\log(35)}{\log(1.23)}
\]

Then:

\[
x = \frac{\log(35)}{\log(1.23)} - 4
\]

### 3. \( 29(1.16)^{x-3} = 87 \)

First divide both sides by 29:

\[
(1.16)^{x-3} = \frac{87}{29}
\]

Now take the logarithm of both sides:

\[
(x-3) \log(1.16) = \log\left(\frac{87}{29}\right)
\]

Solve for \(x\):

\[
x-3 = \frac{\log\left(\frac{87}{29}\right)}{\log(1.16)}
\]

Then:

\[
x = \frac{\log\left(\frac{87}{29}\right)}{\log(1.16)} + 3
\]

### 4. \( 8 + 8(1.12)^{2x} = 24 \)

First, subtract 8 from both sides:

\[
8(1.12)^{2x} = 16
\]

Divide both sides by 8:

\[
(1.12)^{2x} = 2
\]

Now take the logarithm of both sides:

\[
2x \log(1.12) = \log(2)
\]

Solve for \(x\):

\[
x = \frac{\log(2)}{2\log(1.12)}
\]

I will now calculate all values.Here are the solutions to the exponential equations, rounded to three decimal places:

1. \( 8^{x+3} = 22 \) → \( x = -1.514 \)
2. \( 1.23^{x+4} = 35 \) → \( x = 13.174 \)
3. \( 29(1.16)^{x-3} = 87 \) → \( x = 10.402 \)
4. \( 8 + 8(1.12)^{2x} = 24 \) → \( x = 3.058 \)

Let me know if you would like more details on any of these steps.

Here are five related questions to consider:
1. How do you solve exponential equations when both sides have different bases?
2. What are the logarithmic properties useful for solving exponential equations?
3. How can you verify the accuracy of an exponent found through logarithms?
4. How are exponential and logarithmic functions inverses of each other?
5. What happens if the base of the exponential equation is less than 1?

**Tip:** Always double-check your final answers by substituting them back into the original equations to confirm accuracy!

Solve the exponential equations: 8^(x+3) = 22, 1.23^(x+4) = 35, 29(1.16)^(x-3) = 87, 8 + 8(1.12)^(2x) = 24.

Learn how to solve exponential equations such as 8^(x+3) = 22 and 29(1.16)^(x-3) = 87 by applying logarithmic principles. Includes step-by-step solutions and the use of key logarithmic formulas.

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