Solve the equation.
100^(5x+2)=1000^(4x-1)
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To solve the equation 100^(5x+2) = 1000^(4x-1), we can start by taking the logarithm of both sides of the equation. Since both 100 and 1000 are powers of 10, we can use the logarithm base 10, also known as the common logarithm or log.
Using the logarithmic property log(a^b) = b*log(a), we have:
(5x + 2)log(100) = (4x - 1)log(1000)
Recall that log(100) = log(10^2) = 2 and log(1000) = log(10^3) = 3. Substituting these values into the equation, we get:
(5x + 2) * 2 = (4x - 1) * 3
Simplifying further, we have:
10x + 4 = 12x - 3
Now, let's isolate the variables on one side of the equation. Subtracting 10x from both sides, we get:
4 = 2x - 3
Next, we add 3 to both sides:
7 = 2x
Finally, divide both sides by 2:
x = 7/2
Therefore, the solution to the equation 100^(5x+2) = 1000^(4x-1) is x = 7/2.
To solve the equation 100^(5x+2) = 1000^(4x-1), we can simplify both sides of the equation by using the properties of exponents.
First, let's focus on the bases of the exponential terms, which are 100 and 1000.
Since 1000 is equal to 10^3, we can rewrite the right side of the equation as (10^3)^(4x-1). Using the property (a^b)^c = a^(b*c), we can simplify this to 10^(3*(4x-1)), which becomes 10^(12x-3).
Now our equation becomes 100^(5x+2) = 10^(12x-3).
Next, we need to consider the exponents. We know that (a^b)^c = a^(b*c), so we can rewrite 100 as 10^2 and 10 as 10^1 in the left side of the equation:
(10^2)^(5x+2) = 10^(12x-3).
Using the property (a^b)^c = a^(b*c), we can simplify the left side of the equation to 10^(2*(5x+2)), which becomes 10^(10x+4).
Now, our equation becomes 10^(10x+4) = 10^(12x-3).
To solve this equation, we need to set the exponents equal to each other:
10x + 4 = 12x - 3.
Next, we can subtract 10x from both sides of the equation:
4 = 2x - 3.
Then, adding 3 to both sides of the equation, we have:
7 = 2x.
Finally, divide both sides of the equation by 2 to solve for x:
x = 7/2.
Therefore, the solution to the equation 100^(5x+2) = 1000^(4x-1) is x = 7/2.
Recall that 100 is 10^2 and that 1000 is 10^3, so you have power of a power...
that is 10^(2(5x+2)= 10^(3(4x-1) and since the bases are the same we can set the exponents the same so...
2(5x+2)=3(4x-1) and use the distributive property to expand the brackets, then collect like terms and solve for x.