Solve algebraically. a) 4^2x=4^6 b) 5^x=625 c) 3^2x+1=9 d) 10^x+1=10^2x-3 e) 4^3x-2=32^x+1 f) 25^x+1=125^x-2

## Step 1: Solving for x in exponential equations with the same base### When the bases are the same, set the exponents equal to each other and solve for .## Step 2: Solving ### Since the bases are the same, we equate the exponents: . Dividing both sides by 2 gives .## Step 3: Solving ### Rewrite 625 as a power of 5: . Thus, . Equating exponents gives .## Step 4: Solving ### Rewrite 9 as a power of 3: . Thus, . Equating exponents gives . Subtracting 1 from both sides gives . Dividing both sides by 2 gives .## Step 5: Solving ### Since the bases are the same, we equate the exponents: . Subtracting from both sides gives . Adding 3 to both sides gives .## Step 6: Solving ### Rewrite both sides with a common base of 2. and . So, . This simplifies to . Equating exponents gives . Subtracting from both sides gives . Adding 4 to both sides gives .## Step 7: Solving ### Rewrite both sides with a common base of 5. and . So, . This simplifies to . Equating exponents gives . Subtracting from both sides gives . Adding 6 to both sides gives .