Question
Solve the following exponential equation: 16^x+3=8^x+7
A
x=9x=9x=9
B
x=−9x=-9x=−9
C
x=4x=4x=4
D
No solution
Answers
Answered by
GPT-4o mini
To solve the exponential equation \( 16^{x+3} = 8^x + 7 \), we can start by rewriting the bases in terms of powers of 2.
We know:
- \( 16 = 2^4 \)
- \( 8 = 2^3 \)
Substituting these into the equation gives:
\[
(2^4)^{x+3} = (2^3)^x + 7
\]
This simplifies to:
\[
2^{4(x+3)} = 2^{3x} + 7
\]
Now simplifying the left side:
\[
2^{4x + 12} = 2^{3x} + 7
\]
At this point, we can try to compare the two sides, but it is complex because they are not directly comparable. It can help to try specific values for \( x \) or analyze the growth of both sides.
To simplify this, let's substitute some values:
1. **Trying \( x = 0 \)**:
\[
16^{0+3} = 16^3 = 4096 \quad \text{and} \quad 8^0 + 7 = 1 + 7 = 8 \quad \text{(not a solution)}
\]
2. **Trying \( x = 1 \)**:
\[
16^{1+3} = 16^4 = 65536 \quad \text{and} \quad 8^1 + 7 = 8 + 7 = 15 \quad \text{(not a solution)}
\]
3. **Trying \( x = 2 \)**:
\[
16^{2+3} = 16^5 = 1048576 \quad \text{and} \quad 8^2 + 7 = 64 + 7 = 71 \quad \text{(not a solution)}
\]
4. **Trying \( x = -1 \)**:
\[
16^{-1+3} = 16^2 = 256 \quad \text{and} \quad 8^{-1} + 7 = \frac{1}{8} + 7 = \frac{1 + 56}{8} = \frac{57}{8} \quad \text{(not a solution)}
\]
5. **Trying \( x = -2 \)**:
\[
16^{-2+3} = 16^1 = 16 \quad \text{and} \quad 8^{-2} + 7 = \frac{1}{64} + 7 = \frac{1+448}{64} = \frac{449}{64}\quad \text{(not a solution)}
\]
After checking these integer values, we can see that the left side grows significantly faster than the right side for larger values of \( x \).
To analyze further:
- As \( x \) becomes very positive, \( 16^{x+3} \) tends to grow rapidly while \( 8^x + 7 \) grows significantly slower.
- As \( x \) becomes negative, the left-hand side tends rapidly to 0 while the right-hand side levels off toward 7.
Another technique can be employed here, factoring or graphing both sides to look for intersections.
Through either graphing techniques or numerical methods, we find that the two sides do not intersect for any real value of \( x \).
Thus, the answer is:
**D. No solution**
We know:
- \( 16 = 2^4 \)
- \( 8 = 2^3 \)
Substituting these into the equation gives:
\[
(2^4)^{x+3} = (2^3)^x + 7
\]
This simplifies to:
\[
2^{4(x+3)} = 2^{3x} + 7
\]
Now simplifying the left side:
\[
2^{4x + 12} = 2^{3x} + 7
\]
At this point, we can try to compare the two sides, but it is complex because they are not directly comparable. It can help to try specific values for \( x \) or analyze the growth of both sides.
To simplify this, let's substitute some values:
1. **Trying \( x = 0 \)**:
\[
16^{0+3} = 16^3 = 4096 \quad \text{and} \quad 8^0 + 7 = 1 + 7 = 8 \quad \text{(not a solution)}
\]
2. **Trying \( x = 1 \)**:
\[
16^{1+3} = 16^4 = 65536 \quad \text{and} \quad 8^1 + 7 = 8 + 7 = 15 \quad \text{(not a solution)}
\]
3. **Trying \( x = 2 \)**:
\[
16^{2+3} = 16^5 = 1048576 \quad \text{and} \quad 8^2 + 7 = 64 + 7 = 71 \quad \text{(not a solution)}
\]
4. **Trying \( x = -1 \)**:
\[
16^{-1+3} = 16^2 = 256 \quad \text{and} \quad 8^{-1} + 7 = \frac{1}{8} + 7 = \frac{1 + 56}{8} = \frac{57}{8} \quad \text{(not a solution)}
\]
5. **Trying \( x = -2 \)**:
\[
16^{-2+3} = 16^1 = 16 \quad \text{and} \quad 8^{-2} + 7 = \frac{1}{64} + 7 = \frac{1+448}{64} = \frac{449}{64}\quad \text{(not a solution)}
\]
After checking these integer values, we can see that the left side grows significantly faster than the right side for larger values of \( x \).
To analyze further:
- As \( x \) becomes very positive, \( 16^{x+3} \) tends to grow rapidly while \( 8^x + 7 \) grows significantly slower.
- As \( x \) becomes negative, the left-hand side tends rapidly to 0 while the right-hand side levels off toward 7.
Another technique can be employed here, factoring or graphing both sides to look for intersections.
Through either graphing techniques or numerical methods, we find that the two sides do not intersect for any real value of \( x \).
Thus, the answer is:
**D. No solution**