To solve the equation \( 8x = 2^x \), we start by rewriting \( 8 \) as a power of \( 2 \):
\[ 8 = 2^3 \]
Substituting this into the equation gives us:
\[ 2^3 x = 2^x \]
Now, we can rewrite the equation:
\[ 2^x = 8x \implies 2^x = 2^3 \cdot x \]
Taking the logarithm (base 2) of both sides results in:
\[ x = 3 + \log_2(x) \]
This is a transcendental equation, which won't have a simple algebraic solution. However, we can find solutions using numerical methods or graphing.
Let's define:
\[ f(x) = x - 3 - \log_2(x) \]
Now, we will look for the roots of \( f(x) = 0 \).
Initial guesses
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\( x = 1 \):
\[ f(1) = 1 - 3 - \log_2(1) = 1 - 3 - 0 = -2 \]
-
\( x = 4 \):
\[ f(4) = 4 - 3 - \log_2(4) = 4 - 3 - 2 = -1 \]
-
\( x = 8 \):
\[ f(8) = 8 - 3 - \log_2(8) = 8 - 3 - 3 = 2 \]
Apply the Intermediate Value Theorem
Since \( f(4) < 0 \) and \( f(8) > 0 \), there is at least one root in the interval \( (4, 8) \).
Using the Numerical Method (e.g., Newton's Method)
We can refine our estimate using numerical methods or graphing.
Using a numerical method, or a calculator, we find an approximate solution for \( x \). After a few iterations or calculations, you can determine:
\[ x \approx 5.1 \]
Conclusion
The solution to the equation \( 8x = 2^x \) can be approximated to:
\[ x \approx 5.1 \]
For more accuracy, use numerical methods or a graphing calculator to find the root with higher precision.