Solve for x (1/2)^x=1

Solution:

Step 1:

Apply the natural logarithm (ln) to both sides of the equation $(1/2)^x = 1$ to extract the exponent.

$$\ln((1/2)^x) = \ln(1)$$

Step 2:

Utilize the logarithm properties to simplify the equation.

Step 2.1:

Bring down the exponent in front of the ln function.

$$x \cdot \ln(1/2) = \ln(1)$$

Step 2.2:

Express $\ln(1/2)$ as a difference of logarithms.

$$x \cdot (\ln(1) - \ln(2)) = \ln(1)$$

Step 2.3:

Recognize that $\ln(1)$ equals zero.

$$x \cdot (0 - \ln(2)) = \ln(1)$$

Step 2.4:

Simplify the expression by combining terms.

$$x \cdot (-\ln(2)) = \ln(1)$$

Step 3:

Further simplify the left-hand side of the equation.

Step 3.1:

Rearrange the factors, noting the negative sign.

$$-x \cdot \ln(2) = \ln(1)$$

Step 4:

Clarify the right-hand side of the equation.

Step 4.1:

Use the fact that $\ln(1)$ is zero.

$$-x \cdot \ln(2) = 0$$

Step 5:

Isolate the variable $x$ by dividing both sides of the equation by $-\ln(2)$.

Step 5.1:

Perform the division on both sides.

$$\frac{-x \cdot \ln(2)}{-\ln(2)} = \frac{0}{-\ln(2)}$$

Step 5.2:

Simplify the left-hand side by canceling out the common factors.

Step 5.2.1:

Note that dividing two negatives yields a positive.

$$\frac{x \cdot \ln(2)}{\ln(2)} = \frac{0}{-\ln(2)}$$

Step 5.2.2:

Eliminate the common $\ln(2)$ factor.

Step 5.2.2.1:

Cancel out the common terms.

$$\frac{x \cancel{\ln(2)}}{\cancel{\ln(2)}} = \frac{0}{-\ln(2)}$$

Step 5.2.2.2:

Simplify the expression for $x$.

$$x = \frac{0}{-\ln(2)}$$

Step 5.3:

Finalize the right-hand side.

Step 5.3.1:

Zero divided by any non-zero number is zero.

$$x = 0$$

Knowledge Notes:

To solve the equation $(1/2)^x = 1$, we utilize the properties of logarithms. The natural logarithm function, denoted as $\ln$, is the inverse of the exponential function $e^x$. Here are the relevant knowledge points and explanations:

1. Logarithm Properties: The logarithm of a power, $\ln(a^b)$, can be rewritten as $b \cdot \ln(a)$. This property allows us to move the exponent in front of the logarithm, making it easier to solve equations involving exponents.

2. Natural Logarithm of One: The natural logarithm of 1 is always zero, $\ln(1) = 0$. This is because $e^0 = 1$, and since $\ln$ is the inverse of the exponential function, $\ln(1)$ must equal zero.

3. Simplifying Logarithmic Expressions: When simplifying expressions involving logarithms, we can use the properties of logarithms to combine or separate terms. For instance, $\ln(a/b)$ can be written as $\ln(a) - \ln(b)$.

4. Isolating Variables: To solve for a variable, we often need to isolate it on one side of the equation. This can involve using arithmetic operations such as addition, subtraction, multiplication, and division.

5. Zero Divided by a Number: Any time you have zero divided by a non-zero number, the result is zero. This is because zero divided into any number of parts is still zero.

By applying these principles, we can solve the given equation and find that $x = 0$ is the solution.