Solve:
3e2x + 2 = 50
a. ln 8 b. ln 0.25
c. ln 14 d. 0.5ln 16
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Solve:
1 = 7
ex
a. ln 1/7 b. ln 1/7
c. ln 7 d. ln 7
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Solve:
e3x = 8
a. ln 5 b. 3 ln 8
c. ln 8
3
d. ln 3
8
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ln 1
√ e
a. -√e
b. -1
2
c. 1
e
d. 1
2
To solve these equations, we need to isolate the variable x by using the properties of logarithms and exponential functions. Here's how you can solve each of these equations:
1. Solve: 3e^(2x) + 2 = 50
Step 1: Subtract 2 from both sides to isolate the exponential term.
3e^(2x) = 48
Step 2: Divide both sides by 3 to isolate the exponential term.
e^(2x) = 16
Step 3: Take the natural logarithm (ln) of both sides to eliminate the exponential.
ln(e^(2x)) = ln(16)
2x = ln(16)
Step 4: Divide both sides by 2 to solve for x.
x = (1/2)ln(16)
Thus, the answer is (d) 0.5ln(16).
2. Solve: 1 = 7e^x
Step 1: Divide both sides by 7 to isolate the exponential term.
1/7 = e^x
Step 2: Take the natural logarithm (ln) of both sides to eliminate the exponential.
ln(1/7) = ln(e^x)
ln(1/7) = x
Thus, the answer is (a) ln(1/7).
3. Solve: e^(3x) = 8
Step 1: Take the natural logarithm (ln) of both sides to eliminate the exponential.
ln(e^(3x)) = ln(8)
3x = ln(8)
Step 2: Divide both sides by 3 to solve for x.
x = (1/3)ln(8)
Thus, the answer is (b) 3ln(8).
4. Solve: ln(1/√e)
Step 1: Simplify the expression under the natural logarithm.
ln(1/√e) = ln(e^(-1/2))
Step 2: Use the properties of logarithms to rewrite the expression.
ln(e^(-1/2)) = -1/2
Thus, the answer is (a) -√e.