Find x when 5^2 = 125^(2+x).

Answer 1

Before taking the logarithm of both sides, we must have the same base. On the left side, our base if 5, which cannot be reduced further. However, 125=5^3, and we multiply the exponents together. Then, our equation reduces to 2=6+3x, which can be solved to have x=-4/3

Answer 2

So here we have

5^2 is equal to 125 ^ (2 +x) power

We want to get a common base for everything

125 is the same thing as 5 ^3

So we can re write this now as

5^2 = 5 ^3 (2+x)

We are here doing 2 exponents multiplied together so we need to distribute the 3

We now have 5 ^2 = 5 ^ (6+3x)

With a common base, we can now set the exponents equal to each other

This means that 2= 6+ 3x

solve the way you would any other linear equation. Subtract the 6

-4 = 3x

then divide by the 3

-4/3 = x

X is negative 1

Answer 3

5^2 = 125^(2+x)

5^2 = 5^3^(2+x) = 5^(6+3x)

if base is same, exponents are the same

6+3x =2

3x = 2-6 =-4

x = -4/3

check the answer

5^2 = 25 = 125^(2-4/3) = 125^(2/3) = (cube root of 125)^2 = 5^2 = 25

Answer 4

To solve an equation is to find the value of the variable that makes the statement true. The equation

5² = 125^2+x can be solved if we apply a property in mathematics that says: if two powers are equal and they have the same base, then we can conclude that their exponents are equal, like this:

If it´s true that b^m = bⁿ , then m = n.

If we can make the bases to be the same, then the above property can be applied. It´s possible to make them the same since 125 = 5³. Let´s substitute this value and the equation now will look like this:

5²= (5³)^2+x

Another property of powers states that the power of a power, we can multiply the exponents. Now the equation can look like this:

5²=5^6+3x (we multiplied the exponents). Now that both powers have the same base we can conclude that their exponents are also equal, so

2 = 6+3x equation that can be easily solved by using inverse operations. We get

-4 = 3x by subtracting 6 on both sides. We get

-4/3 = x by dividing by 3 on both sides.

x = -4/3

Answer 5

See the videa.