Questions: Solve the exponential equation. 10^(-x) = 5^(2x) Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. x= (Type an integer or a decimal. Do not round until the final answer. Then round to four decimal B. The solution is not a real number.

Solve the exponential equation.
10^(-x) = 5^(2x)

Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A. x=
(Type an integer or a decimal. Do not round until the final answer. Then round to four decimal
B. The solution is not a real number.

Transcript text: Solve the exponential equation. \[ 10^{-x}=5^{2 x} \] Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. $x=$ $\square$ (Type an integer or a decimal. Do not round until the final answer. Then round to four decimal B. The solution is not a real number.

Solution

Solution Steps

To solve the exponential equation $10^{-x} = 5^{2x}$, we can take the logarithm of both sides to simplify the equation. By using properties of logarithms, we can solve for $x$.

Step 1: Take the Logarithm of Both Sides

We start with the equation: \[ 10^{-x} = 5^{2x} \] Taking the logarithm of both sides gives us: \[ \log(10^{-x}) = \log(5^{2x}) \]

Step 2: Apply Logarithmic Properties

Using the properties of logarithms, we can rewrite the equation: \[ -x \log(10) = 2x \log(5) \]

Step 3: Rearrange the Equation

Rearranging the equation leads to: \[ -x \log(10) - 2x \log(5) = 0 \] Factoring out $x$ gives: \[ x(-\log(10) - 2 \log(5)) = 0 \] This implies either $x = 0$ or: \[ -\log(10) - 2 \log(5) = 0 \]

Step 4: Solve for $x$

To find the non-trivial solution, we solve: \[ x = -\frac{\log(10)}{2 \log(5)} \] Calculating this yields: \[ x \approx -0.7153 \]