Solve the following exponential equations algebraically. Show in
detail your workings and do the checks in detail to get full marks.
√5 = 5^𝑥+4
To solve the equation √5 = 5^(x+4) algebraically, we need to isolate the variable x.
Step 1: Rewrite the equation with the bases of both sides of the equation the same. We can rewrite √5 as 5^(1/2) since the square root of a number is the same as the number raised to the power of 1/2.
So, the equation becomes: 5^(1/2) = 5^(x+4).
Step 2: Since the bases are the same, we can set the exponents equal to each other.
1/2 = x + 4.
Step 3: Now, solve for x.
x = 1/2 - 4
x = -7/2
Step 4: Check the solution by substituting x back into the original equation.
√5 = 5^(-7/2 + 4)
√5 = 5^(1/2)
√5 = √5
The solution x = -7/2 satisfies the original equation.
Therefore, the solution to the equation √5 = 5^(x+4) is x = -7/2.