Question

Solve each equation. Express irrational solutions in exact form. $$ \log _{2} x^{\log _{2} x}=4 $$

Short answer

The solutions are \ \[ x = 4 \] and \ \[ x = \frac{1}{4} \]

Step-by-step solution

Recognize the logarithmic equation

The given equation is \ \ \ \ \[ \log _{2} x^{\log _{2} x}=4 \] the goal will be to simplify and solve for x.

Simplify the exponent using properties of logarithms

Notice that the left side of the equation can be rewritten using the property of logarithms. Specifically, \ \ \[ \log_{a} (b^c) = c \cdot \log_{a} b \] \ then, applying this to the LHS of our equation: \[ \log_{2} x^{\log_{2} x} = \log_{2} x \cdot \log_{2} x = (\log_{2} x)^2 \] Now the equation becomes \ \[ (\log_{2} x)^2 = 4 \]

Solve for \ \log_{2} x

Take the square root of both sides to solve for \ \log_{2} x. \ \[ \sqrt{(\log_{2} x)^2} = \sqrt{4} \] This simplifies to: \ \[ \log_{2} x = \pm 2 \]

Solve for x

Solve for x by converting the logarithmic form to its exponential form. For \ \[ \log_{2} x = 2 \] convert it to \ \[ x = 2^2 = 4 \] Similarly, for \ \[ \log_{2} x = -2 \] convert it to \[ x = 2^{-2} = \frac{1}{4} \]

Key concepts

properties of logarithms

To solve logarithmic equations effectively, it's crucial to understand the properties of logarithms. These properties simplify logarithmic expressions, making it easier to manipulate and solve them. The main properties are:

• Product Property: \ \( \log_{a}(bc) = \log_{a}(b) + \log_{a}(c) \ \)
• Quotient Property: \ \( \log_{a}( \frac{b}{c}) = \log_{a}(b) - \log_{a}(c) \ \)
• Power Property: \ \( \log_{a}(b^c) = c \cdot \log_{a}(b) \ \)

These rules help simplify complex logarithmic equations.
For instance, in the given exercise \ \( \log_{2}(x^{\log_{2}(x)}) = 4 \ \), the power property transforms \ \( x^{\log_{2}(x)} \ \) into \ \( \log_{2}(x) \cdot \log_{2}(x) = \log_{2}(x)^2 \ \). This simplification is a critical step in solving the equation.

solving logarithmic equations

Solving logarithmic equations requires a step-by-step approach. Here's how to do it:

• Step 1: Simplify the equation using properties of logarithms. In our exercise, we used the power property to rewrite \ \( \log_{2}(x^{\log_{2}(x)}) = 4 \ \) as \ \( \log_{2}(x)^2 = 4 \ \).
• Step 2: Isolate the logarithmic term. We got \ \( (\log_{2}(x))^2 = 4 \ \).
• Step 3: Solve for the logarithmic term. Taking the square root on both sides, we find \ \( \log_{2}(x) = \pm 2 \ \).
• Step 4: Convert from logarithmic to exponential form. For example, \ \( \log_{2}(x) = 2 \ \) becomes \ \( x = 2^2 = 4 \ \), and \ \( \log_{2}(x) = -2 \ \) becomes \ \( x = 2^{-2} = \frac{1}{4} \ \).

With these steps, you can solve many logarithmic equations.

exponential form

After handling the logarithmic equation, you may need to convert logarithms to exponential form. This makes solutions explicit and easier to understand.
To convert a logarithmic statement \ \( y = \log_{a}(x) \ \) to exponential form, rewrite it as \ \( x = a^y \ \). This is key for solving the original equation:

• For \ \( \log_{2}(x) = 2 \ \), convert to \ \( x = 2^2 = 4 \ \).
• For \ \( \log_{2}(x) = -2 \ \), convert to \ \( x = 2^{-2} = \frac{1}{4} \ \).

Understanding this conversion helps bridge logarithmic equations and their solutions.