Answer :
Sure! Let's solve each part of the question step-by-step.
(c) Solve the equation [tex]\(4^{h(x)} = 8\)[/tex].
1. Change the bases to be the same:
- Notice that 4 and 8 can both be expressed as powers of 2:
- [tex]\(4 = 2^2\)[/tex]
- [tex]\(8 = 2^3\)[/tex]
2. Rewrite the equation:
- Substitute the equivalent powers of 2 into the original equation:
- [tex]\((2^2)^{h(x)} = 2^3\)[/tex]
3. Simplify the equation:
- Apply the power of a power property [tex]\((a^m)^n = a^{m \cdot n}\)[/tex]:
- [tex]\(2^{2h(x)} = 2^3\)[/tex]
4. Solve for [tex]\(h(x)\)[/tex]:
- Since the bases are the same, set the exponents equal to each other:
- [tex]\(2h(x) = 3\)[/tex]
5. Find [tex]\(h(x)\)[/tex]:
- Divide both sides by 2:
- [tex]\(h(x) = \frac{3}{2}\)[/tex]
Answer: [tex]\(h(x) = \frac{3}{2}\)[/tex]
---
(d) Solve the inequality [tex]\(l(x) = \frac{1}{2}\)[/tex].
1. Understanding [tex]\(D_l\)[/tex]:
- [tex]\(D_l\)[/tex] would generally represent the domain of the function [tex]\(l(x)\)[/tex]. Without specific information about [tex]\(l(x)\)[/tex], we proceed by assuming it refers to all possible values that [tex]\(l(x)\)[/tex] can take.
2. Solve [tex]\(l(x) = \frac{1}{2}\)[/tex]:
- If we assume a simple scenario where [tex]\(l(x) = x\)[/tex], then solving [tex]\(x = \frac{1}{2}\)[/tex] gives a solution directly:
- [tex]\(x = \frac{1}{2}\)[/tex]
Answer: [tex]\(x = \frac{1}{2}\)[/tex]
These solutions are based on a typical scenario for solving equations and inequalities with the provided expressions. If the function [tex]\(l(x)\)[/tex] is different, the same method applies once the form of [tex]\(l(x)\)[/tex] is known.
(c) Solve the equation [tex]\(4^{h(x)} = 8\)[/tex].
1. Change the bases to be the same:
- Notice that 4 and 8 can both be expressed as powers of 2:
- [tex]\(4 = 2^2\)[/tex]
- [tex]\(8 = 2^3\)[/tex]
2. Rewrite the equation:
- Substitute the equivalent powers of 2 into the original equation:
- [tex]\((2^2)^{h(x)} = 2^3\)[/tex]
3. Simplify the equation:
- Apply the power of a power property [tex]\((a^m)^n = a^{m \cdot n}\)[/tex]:
- [tex]\(2^{2h(x)} = 2^3\)[/tex]
4. Solve for [tex]\(h(x)\)[/tex]:
- Since the bases are the same, set the exponents equal to each other:
- [tex]\(2h(x) = 3\)[/tex]
5. Find [tex]\(h(x)\)[/tex]:
- Divide both sides by 2:
- [tex]\(h(x) = \frac{3}{2}\)[/tex]
Answer: [tex]\(h(x) = \frac{3}{2}\)[/tex]
---
(d) Solve the inequality [tex]\(l(x) = \frac{1}{2}\)[/tex].
1. Understanding [tex]\(D_l\)[/tex]:
- [tex]\(D_l\)[/tex] would generally represent the domain of the function [tex]\(l(x)\)[/tex]. Without specific information about [tex]\(l(x)\)[/tex], we proceed by assuming it refers to all possible values that [tex]\(l(x)\)[/tex] can take.
2. Solve [tex]\(l(x) = \frac{1}{2}\)[/tex]:
- If we assume a simple scenario where [tex]\(l(x) = x\)[/tex], then solving [tex]\(x = \frac{1}{2}\)[/tex] gives a solution directly:
- [tex]\(x = \frac{1}{2}\)[/tex]
Answer: [tex]\(x = \frac{1}{2}\)[/tex]
These solutions are based on a typical scenario for solving equations and inequalities with the provided expressions. If the function [tex]\(l(x)\)[/tex] is different, the same method applies once the form of [tex]\(l(x)\)[/tex] is known.
