What is the value of x in the equation log_x(2401) = 4?
Understand the Problem
The question is asking to find the value of $x$ in the logarithmic equation $log_x(2401) = 4$, which means $x^4 = 2401$. We will solve for $x$ by recognizing that $2401$ can be expressed as a power of a base.
Answer
The solution is $7$.
Answer for screen readers
The value of $x$ is $7$.
Steps to Solve
- Understanding the equation
We start with the equation given by the logarithmic expression:
$$ log_x(2401) = 4 $$
This means that:
$$ x^4 = 2401 $$
- Finding the prime factorization of 2401
Next, we need to express $2401$ as a power. We can check the prime factorization of $2401$:
$$ 2401 = 7 \times 7 \times 7 \times 7 = 7^4 $$
- Equating the powers
Now we can equate the powers:
$$ x^4 = 7^4 $$
- Solving for x
By taking the fourth root of both sides, we can solve for $x$:
$$ x = 7 $$
The value of $x$ is $7$.
More Information
The number $2401$ is significant as it is a perfect power and can be expressed as $7^4$. This shows the connection between logarithmic functions and exponents.
Tips
- A common mistake is to misinterpret the logarithmic equation. It's important to correctly translate the logarithmic relationship into an exponential form.
- Another mistake is forgetting to consider both positive and negative roots when taking an even root, although in logarithmic bases, we only accept positive values.
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