Question

Solve the equation log₄(xˡᵒᵍ⁴⁽ˣ⁾)=9 for x. What are the solutions?

Answer 1

`\[ x = 4^9 \]`

To solve the equation
`\( \log_4(x^(\log_4(x))) = 9 \)` for x, the solution is
`\( x = 4^9 \)`.

Step-by-step explanation:

The given equation is a logarithmic equation, and to solve for x, we use the property that
`\( \log_a(b^c) = c \cdot \log_a(b) \)`. Applying this property to the equation, we get
`\( \log_4(x \log_4(x)) = 9`. Now, we can rewrite the equation using the base 4 logarithm:
`\( \log_4(x) + \log_4(\log_4(x)) = 9 \)`.

To simplify further, let ( y =
`log_4`(x) ). The equation becomes
`\( y + log_4(y) = 9`. Solving this equation for y, we find ( y = 3 ). Now, substitute (
`log_4`(x) = 3 ) back into the original variable, and we get \( x = 4^3 \). Therefore, the solution is ( x =
`4^9` ).

Logarithmic equations involve manipulating logarithmic properties to isolate the variable. Understanding the rules of logarithms and exponentials is crucial in solving such equations effectively.