Solve this indical equation:

Question

(10)^(x) = 0.001

Answer 1

To solve the exponential equation \( (10)^x = 0.001 \), we can first rewrite \( 0.001 \) in terms of a power of \( 10 \):

\[ 0.001 = \frac{1}{1000} = \frac{1}{10^3} = 10^{-3} \]

Now, we can rewrite the equation as:

\[ 10^x = 10^{-3} \]

Since the bases are the same, we can set the exponents equal to each other:

\[ x = -3 \]

Thus, the solution to the equation \( (10)^x = 0.001 \) is:

\[ \boxed{-3} \]

Answer 2

(5)^(1-2x) = 1

Answer 3

To solve the equation \( (5)^{1 - 2x} = 1 \), we start by recalling that any number raised to the power of 0 equals 1. Therefore, we can set the exponent equal to 0:

\[ 1 - 2x = 0 \]

Now, solve for \( x \):

\[ 1 = 2x \]

\[ x = \frac{1}{2} \]

Thus, the solution to the equation \( (5)^{1 - 2x} = 1 \) is:

\[ \boxed{\frac{1}{2}} \]

Answer 4

Can you like do your work together with the base instead of separate calculations of the indices and base?

Answer 5

Sure! Let's solve the equation \( (5)^{1 - 2x} = 1 \) while keeping the base and the exponent together for clarity.

1. We start with the equation:

   \[ (5)^{1 - 2x} = 1 \]

2. We know that for any base \( a \) (where \( a \neq 0 \)), \( a^0 = 1 \). Therefore, we can set the exponent equal to 0:

   \[ 1 - 2x = 0 \]

3. Solving for \( x \):

   \[ 1 - 2x = 0 \]

   \[ 1 = 2x \]

   \[ x = \frac{1}{2} \]

Thus, the final solution is:

\[ \boxed{\frac{1}{2}} \]

If you have any other specific approach in mind or need further clarification, feel free to ask!

Answer 6

But what happened to the numbers 5 and 1, the bases?

Answer 7

I understand your request now. Let's take into account the base and keep the calculations together. We'll analyze it with emphasis on the base \(5\) and the exponent.

Given the equation:

\[ (5)^{1 - 2x} = 1 \]

1. Recognize the condition of the equation: Since \( (5)^{1 - 2x} \) must equal 1, we can apply the property that states any non-zero base raised to the power of 0 equals 1. Therefore, we can conclude:

   \[ 1 - 2x = 0 \]

2. Now solve for \(x\) while keeping the equation balanced and highlighting the base and exponent:

   • Start with the equation for the exponent: \[ 1 - 2x = 0 \]

   • Rearranging this gives us: \[ 1 = 2x \]

   • Solving for \(x\): \[ x = \frac{1}{2} \]

Thus, highlighting the significance of the base \(5\) alongside the exponent process, we arrive at the solution:

\[ \boxed{\frac{1}{2}} \]

If you were looking for something else in terms of formatting or explanation, please let me know!