Solve the equation (for x): log1-3-log1-2-0.5 = 0. 2. Solve the equation (for x): log3 (38) 2-x. 3. List the first five terms of the sequence a₁ = 3, an+1 an n nEN.
The Correct Answer and Explanation is :
Let’s break each part of the problem into steps to solve:
1. Solve the equation:
[
\log_{1-3} – \log_{1-2} – 0.5 = 0
]
This equation is unclear because the logarithmic base ((\log_{1-b})) is unconventional and may not follow typical logarithmic rules since (1-b \leq 0) for most (b). Logarithms are only defined for positive bases not equal to 1. Therefore, this equation cannot be solved as written. Please clarify the equation (perhaps (\log_{a}(b)) or (\ln)) for further explanation.
2. Solve the equation:
[
\log_3(38) = 2 – x
]
Rewriting this equation:
[
x = 2 – \log_3(38)
]
Using the change of base formula ((\log_a(b) = \frac{\ln(b)}{\ln(a)})):
[
\log_3(38) = \frac{\ln(38)}{\ln(3)}
]
Thus,
[
x = 2 – \frac{\ln(38)}{\ln(3)}
]
Numerical approximation:
[
\ln(38) \approx 3.6376, \quad \ln(3) \approx 1.0986
]
[
x = 2 – \frac{3.6376}{1.0986} \approx 2 – 3.31 = -1.31
]
3. List the first five terms of the sequence
The sequence is defined as:
[
a_1 = 3, \quad a_{n+1} = a_n \cdot n
]
First five terms:
- (a_1 = 3)
- (a_2 = a_1 \cdot 1 = 3 \cdot 1 = 3)
- (a_3 = a_2 \cdot 2 = 3 \cdot 2 = 6)
- (a_4 = a_3 \cdot 3 = 6 \cdot 3 = 18)
- (a_5 = a_4 \cdot 4 = 18 \cdot 4 = 72)
The first five terms are:
[
3, 3, 6, 18, 72
]
Explanation (300 words):
The above problems focus on solving logarithmic equations and understanding recursive sequences.
For the second equation, logarithms help us express exponential relationships. Using the change of base formula, we convert (\log_3(38)) into a ratio of natural logarithms. After computing this value, we substitute it back to solve for (x). The change of base formula is a powerful tool when solving logarithmic equations involving uncommon bases.
For the sequence problem, we use a recursive formula where each term depends on the previous term and the index (n). Starting with the initial value (a_1 = 3), we calculate successive terms by multiplying the prior term by (n). Recursive sequences are widely used in mathematical modeling and algorithms.
If there are any clarifications or additional details required for part 1, please provide further information!