53.)
49x-2 = 7Ö 7
a. -5/4
b.5/4
c. 11/4
d. -11/4
79.)
10
E 3n-2
n=4
a. 14+17+20+...+32
b. 28+31+34+...+43
c. 10+13+16+...+28
d. 9+14+17+...+30
66.
ln 1
√e
a. -√e
b. -1/2
c. 1/e
d. 1/2
67.
Solve:
e3x = 8
a. ln 5
b. 3 ln 8
c. ln 8
3
d. ln 3
8
53.)
To solve the equation 49x-2 = 7√7, we can follow these steps:
1. Add 2 to both sides of the equation: 49x = 7√7 + 2.
2. Divide both sides of the equation by 49: x = (7√7 + 2)/49.
3. Simplify the right side of the equation: x = (7√7 + 2)/49.
4. Rationalize the denominator by multiplying both the numerator and denominator by √7:
x = (7√7 + 2/49)(√7/√7)
x = (7√7 + 2√7)/49√7
x = (9√7)/49√7
x = 9/49
Therefore, the answer is option a) -5/4.
79.)
To find the value of 10 * E (3n-2) for n=4, we can substitute the value of n into the expression and evaluate it step by step:
1. Start by substituting n=4 in the expression: 10 * E (3(4)-2)
Simplify inside the parentheses: 10 * E (12-2)
Simplify further: 10 * E (10)
2. Since the expression E (10) represents a sequence, we need to find the sum of the sequence.
The formula to find the sum of an arithmetic sequence is: Sn = (n/2)(a + l), where n is the number of terms, a is the first term, and l is the last term.
In this case, the first term (a) is 10, and we need to figure out the last term (l) using the given condition.
The last term can be found by substituting n=4 in the expression 3n-2:
l = 3(4) - 2 = 12 - 2 = 10.
Now, substituting the found values into the formula: Sn = (4/2)(10 + 10)
Simplify: Sn = 2 * 20
Sn = 40
Therefore, the answer is option c) 10 + 13 + 16 + ... + 28.
66.)
To simplify ln(1) √e, we can follow these steps:
1. Start by evaluating ln(1). The natural logarithm of 1 is always 0. So, ln(1) = 0.
2. Next, simplify √e. The square root of e (√e) is a constant value (approximately 1.6487212707).
3. Now, we multiply ln(1) by √e: 0 * √e = 0.
Therefore, the answer is option b) -1/2.
67.)
To solve the equation e^(3x) = 8, we can follow these steps:
1. Take the natural logarithm (ln) of both sides of the equation to get rid of the exponent: ln(e^(3x)) = ln(8).
2. Use the property of logarithms that states ln(a^b) = b * ln(a): 3x * ln(e) = ln(8).
3. Since ln(e) is equal to 1, the equation simplifies to: 3x = ln(8).
4. Divide both sides of the equation by 3: x = ln(8)/3.
Therefore, the answer is option c) ln(8)/3.