Math

Question Solve the exponential equation 25x2=1525^{x-2}=\frac{1}{5} for the unknown variable xx.

Studdy Solution

STEP 1

Assumptions
1. The base of the exponent on the left side of the equation is 25.
2. The exponent is x2x-2.
3. The right side of the equation is 15\frac{1}{5}.

STEP 2

First, we need to express both sides of the equation with the same base. The number 25 can be expressed as 525^2 and the number 15\frac{1}{5} can be expressed as 515^{-1}.
25x2=1525^{x-2}=\frac{1}{5}
(52)x2=51\Rightarrow (5^2)^{x-2}=5^{-1}

STEP 3

Now, apply the power of a power rule in exponentiation. This rule states that (am)n=amn(a^m)^n = a^{mn}.
(52)x2=51(5^2)^{x-2}=5^{-1}
52(x2)=51\Rightarrow 5^{2(x-2)}=5^{-1}

STEP 4

In an equation, if the bases are the same, then the exponents must be equal. So, we can set 2(x2)2(x-2) equal to 1-1.
2(x2)=12(x-2)=-1

STEP 5

Solve the equation for xx by first distributing the 2 on the left side of the equation.
2x4=12x - 4 = -1

STEP 6

Then, add 4 to both sides of the equation to isolate 2x2x on the left side.
2x4+4=1+42x - 4 + 4 = -1 + 4
2x=32x = 3

STEP 7

Finally, divide both sides of the equation by 2 to solve for xx.
x=32x = \frac{3}{2}
So, x=32x = \frac{3}{2} is the solution to the equation 25x2=1525^{x-2}=\frac{1}{5}.

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