10) Solve each equation algebraically. b) 2^2x+1=128 c) 4^3x=8^x-2 a) 2^x=16

## Step 1: Solve for $x$ in $2^x = 16$<br />### Express 16 as a power of 2: $16 = 2^4$. Therefore, $2^x = 2^4$. Since the bases are equal, the exponents must be equal.<br />## Step 2: Equate the exponents<br />### Set $x = 4$.<br /><br />## Step 3: Solve for $x$ in $2^{2x+1} = 128$<br />### Express 128 as a power of 2: $128 = 2^7$. Therefore, $2^{2x+1} = 2^7$. Since the bases are equal, the exponents must be equal.<br /><br />## Step 4: Equate the exponents<br />### Set $2x + 1 = 7$. Subtract 1 from both sides: $2x = 6$. Divide both sides by 2: $x = 3$.<br /><br />## Step 5: Solve for $x$ in $4^{3x} = 8^{x-2}$<br />### Express both sides as powers of 2: $4^{3x} = (2^2)^{3x} = 2^{6x}$ and $8^{x-2} = (2^3)^{x-2} = 2^{3(x-2)} = 2^{3x-6}$. Therefore, $2^{6x} = 2^{3x-6}$.<br /><br />## Step 6: Equate the exponents<br />### Set $6x = 3x - 6$. Subtract $3x$ from both sides: $3x = -6$. Divide both sides by 3: $x = -2$.