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

## Step 1: Solving $2^x = 16$<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: Solution for (a)<br />### Thus, $x = 4$.<br /><br />## Step 3: Solving $2^{2x+1} = 128$<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. So, $2x + 1 = 7$.<br />## Step 4: Solution for (b)<br />### Subtract 1 from both sides: $2x = 6$. Divide both sides by 2: $x = 3$.<br /><br />## Step 5: Solving $4^{3x} = 8^{x-2}$<br />### Express both 4 and 8 as powers of 2: $4 = 2^2$ and $8 = 2^3$. Substitute these into the equation: $(2^2)^{3x} = (2^3)^{x-2}$. Simplify using the power of a power rule: $2^{6x} = 2^{3(x-2)}$.<br />## Step 6: Solution for (c)<br />### Since the bases are equal, the exponents must be equal: $6x = 3(x-2)$. Distribute the 3 on the right side: $6x = 3x - 6$. Subtract $3x$ from both sides: $3x = -6$. Divide both sides by 3: $x = -2$.