Solve the following equations for x: (ग) 2^x+1 + 2^x+2 + 2^x+3 = 448 (ड) 4^x+1 - 8 x 4^x-1 = 32 (ढ) 3^x+2 + 3^x+1 + 2 x 3^x = 126 (फ) 8^x-1 - 23x-2 + 8 = 0

Here are the solutions for each equation:

Equation (ग): $2^{x+1} + 2^{x+2} + 2^{x+3} = 448$

Step 1: Rewrite the terms using the exponent rule $a^{m+n} = a^m \cdot a^n$. $2^x \cdot 2^1 + 2^x \cdot 2^2 + 2^x \cdot 2^3 = 448$ $2 \cdot 2^x + 4 \cdot 2^x + 8 \cdot 2^x = 448$

Step 2: Factor out $2^x$. $2^x (2 + 4 + 8) = 448$ $2^x (14) = 448$

Step 3: Isolate $2^x$. $2^x = \frac{448}{14}$ $2^x = 32$

Step 4: Express 32 as a power of 2. $2^x = 2^5$

Step 5: Equate the exponents. $x = 5$ The solution for (ग) is $\boxed{*x=5*}$.

Equation (ड): $4^{x+1} - 8 \times 4^{x-1} = 32$

Step 1: Rewrite the terms using the exponent rules $a^{m+n} = a^m \cdot a^n$ and $a^{m-n} = a^m \cdot a^{-n}$. $4^x \cdot 4^1 - 8 \cdot 4^x \cdot 4^{-1} = 32$ $4 \cdot 4^x - 8 \cdot 4^x \cdot \frac{1}{4} = 32$ $4 \cdot 4^x - 2 \cdot 4^x = 32$

Step 2: Factor out $4^x$. $4^x (4 - 2) = 32$ $4^x (2) = 32$

Step 3: Isolate $4^x$. $4^x = \frac{32}{2}$ $4^x = 16$

Step 4: Express 16 as a power of 4. $4^x = 4^2$

Step 5: Equate the exponents. $x = 2$ The solution for (ड) is $\boxed{*x=2*}$.

Equation (छ): $3^{x+2} + 3^{x+1} + 2 \times 3^x = 126$

Step 1: Rewrite the terms using the exponent rule $a^{m+n} = a^m \cdot a^n$. $3^x \cdot 3^2 + 3^x \cdot 3^1 + 2 \cdot 3^x = 126$ $9 \cdot 3^x + 3 \cdot 3^x + 2 \cdot 3^x = 126$

Step 2: Factor out $3^x$. $3^x (9 + 3 + 2) = 126$ $3^x (14) = 126$

Step 3: Isolate $3^x$. $3^x = \frac{126}{14}$ $3^x = 9$

Step 4: Express 9 as a power of 3. $3^x = 3^2$

Step 5: Equate the exponents. $x = 2$ The solution for (छ) is $\boxed{*x=2*}$.

Equation (फ): The equation is $8^{x-1} - 23^{x-2} + 8 = 0$. Given the context of the other problems, it is highly probable that the term $23^{x-2}$ is actually $23 \times 8^{x-2}$. We will proceed with this assumption.

Assume the equation is: $8^{x-1} - 23 \times 8^{x-2} + 8 = 0$

Step 1: Rewrite the terms using the exponent rule $a^{m-n} = a^m \cdot a^{-n}$. $8^x \cdot 8^{-1} - 23 \cdot 8^x \cdot 8^{-2} + 8 = 0$ $\frac{8^x}{8} - 23 \cdot \frac{8^x}{64} + 8 = 0$

Step 2: Let $y = 8^x$. Substitute $y$ into the equation. $\frac{y}{8} - \frac{23y}{64} + 8 = 0$

Step 3: Multiply the entire equation by 64 to eliminate the denominators. $64 \left( \frac{y}{8} \right) - 64 \left( \frac{23y}{64} \right) + 64(8) = 0$ $8y - 23y + 512 = 0$

Step 4: Combine like terms. $-15y + 512 = 0$

Step 5: Solve for $y$. $-15y = -512$ $y = \frac{512}{15}$

Step 6: Substitute back $y = 8^x$. $8^x = \frac{512}{15}$

Step 7: Solve for $x$ using logarithms. $x = \log_8 \left( \frac{512}{15} \right)$ Since $512 = 8^3$, we can write: $x = \log_8 (8^3) - \log_8 (15)$ $x = 3 - \log_8 (15)$ The solution for (फ) is $\boxed{*x = 3 - \log_8 (15)*}$.