Answer:
\(x=1\)
Step-by-step explanation:
We are solving for x in the exponential equation:
\(100^x = \left(\!\dfrac{1}{10}\!\right)^{\!(x-3)}\)
First, we can rewrite both exponent bases in terms of 10:
- \(100 = 10^{2}\)
- \(1/10 = 10^{-1}\)
↓↓↓
\((10^2)^x = (10^{-1})^{\!(x-3)}\)
Next, we can apply exponent rules to simplify both sides:
\(10^{2x} = 10^{-(x-3)}\)
\(10^{2x} = 10^{(3-x)}\)
Now that both sides have a base of 10, we can take the common log of both sides:
\(\log(10^{2x}) = \log\!\left(10^{(3-x)}\right)\)
\(2x = 3-x\)
The equation now just requires algebraic manipulation to solve for x:
\(3x=3\)
\(\boxed{x=1}\)
