To solve the equation [tex]\( 4^{2x-3} = 7^x + 2^{x+2} \)[/tex], let's break it down step by step:
1. Rewrite the bases with exponents:
[tex]\[
4^{2x-3} = (2^2)^{2x-3} = 2^{4x-6}
\][/tex]
Similarly, for the right-hand side:
[tex]\[
2^{x+2} = 2^x \cdot 2^2 = 4 \cdot 2^x
\][/tex]
So the equation becomes:
[tex]\[
2^{4x-6} = 7^x + 4 \cdot 2^x
\][/tex]
2. Analyze possible solutions:
This is a transcendental equation, which means it typically does not have a straightforward analytical solution like algebraic equations. Instead, we often need numerical methods to find approximate solutions or argue about possible solutions by substituting feasible numbers.
3. Substitute potential solutions:
Let's test a few values of [tex]\( x \)[/tex] to see if we can find a solution:
- Test [tex]\( x = 0 \)[/tex]:
[tex]\[
2^{4(0)-6} = 2^{-6} = \frac{1}{2^6} = \frac{1}{64}
\][/tex]
[tex]\[
7^0 + 4 \cdot 2^0 = 1 + 4 = 5
\][/tex]
Clearly, [tex]\( \frac{1}{64} \ne 5 \)[/tex].
- Test [tex]\( x = 1 \)[/tex]:
[tex]\[
2^{4(1)-6} = 2^{-2} = \frac{1}{2^2} = \frac{1}{4}
\][/tex]
[tex]\[
7^1 + 4 \cdot 2^1 = 7 + 8 = 15
\][/tex]
Again, [tex]\( \frac{1}{4} \ne 15 \)[/tex].
- Test [tex]\( x = 2 \)[/tex]:
[tex]\[
2^{4(2)-6} = 2^2 = 4
\][/tex]
[tex]\[
7^2 + 4 \cdot 2^2 = 49 + 16 = 65
\][/tex]
[tex]\( 4 \ne 65 \)[/tex].
- Test [tex]\( x = 3 \)[/tex]:
[tex]\[
2^{4(3)-6} = 2^{12-6} = 2^6 = 64
\][/tex]
[tex]\[
7^3 + 4 \cdot 2^3 = 343 + 4 \cdot 8 = 343 + 32 = 375
\][/tex]
[tex]\( 64 \ne 375 \)[/tex].
Given that simple values like 0, 1, 2, and 3 do not solve the equation, and the nature of the equation, we should generally expect that a transcendental equation like this will be best approached either graphically or with numerical methods (e.g., using a computer algebra system, iterative methods, or other numerical techniques).
Let's conclude by offering the observation that solving [tex]\( 2^{4x-6} = 7^x + 4 \cdot 2^x \)[/tex] exactly requires numerical methods due to its complexity. Graphically plotting [tex]\( y = 2^{4x-6} \)[/tex] and [tex]\( y = 7^x + 4 \cdot 2^x \)[/tex] would provide the intersection points, which provide the solutions for [tex]\( x \)[/tex]. From the attempts above, no simple analytical solution exists.
