Question

solve the following elementary exponential equation.
1000^{-x} = 10^{4x - 14}
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4 points
x =

Explanation:

Step1: Express 1000 as a power of 10

Since \(1000 = 10^3\), we can rewrite the left - hand side of the equation \(1000^{-x}\) as \((10^3)^{-x}\).
According to the power - of - a - power rule \((a^m)^n=a^{mn}\), \((10^3)^{-x}=10^{3\times(-x)} = 10^{-3x}\). So the equation \(1000^{-x}=10^{4x - 14}\) becomes \(10^{-3x}=10^{4x-14}\).

Step2: Set the exponents equal

For the exponential equation \(a^m=a^n\) (where \(a>0,a
eq1\)), we have \(m = n\).
Since the base \(a = 10>0,a
eq1\) in the equation \(10^{-3x}=10^{4x - 14}\), we can set the exponents equal to each other: \(-3x=4x - 14\).

Step3: Solve for x

First, add \(3x\) to both sides of the equation \(-3x=4x - 14\):
\(-3x + 3x=4x+3x - 14\), which simplifies to \(0 = 7x-14\).
Then, add 14 to both sides: \(0 + 14=7x-14 + 14\), so \(14 = 7x\).
Finally, divide both sides by 7: \(x=\frac{14}{7}=2\).

Answer:

\(x = 2\)