Solve each of the following equations, giving each answer to four decimal places.
Problem 1 :
3^{x} = 17
Solution :
3^{x} = 17
When we solve an exponential equation, we have to get the same base on both sides. Because, if we have the same base, we can equate the exponents and solve for x. In the given equation, since the bases 3 and 17 are not the powers of same number. we can not convert them to the same base. So, we have to take logarithm on both sides and solve for x.
Take logarithm with base 10 on both sides.
log 3^{x} = log 17
Use the power rule of logarithm.
xlog 3 = log 17
Divide both sides by log 3.
x = 2.5789
Problem 2 :
6^{x} = 99
Solution :
6^{x} = 99
Take logarithm with base 10 on both sides.
log 6^{x} = log 99
Use the power rule of logarithm.
x ⋅ log 6 = log 99
Divide both sides by log 6.
x = 2.5646
Problem 3 :
5 ⋅ 18^{6x} = 26
Solution :
5 ⋅ 18^{6x} = 26
Divide both sides by 5.
18^{6x} = 5.2
Take logarithm with base 10 on both sides.
log 18^{6x} = log 5.2
Use the power rule of logarithm.
6x ⋅ log 18 = log 5.2
Divide both sides by 6log 18.
x = 0.0951
Problem 4 :
e^{x - 1 } = 10
Solution :
e^{x - 1 } = 10
Since the base on the left side is e, take natural logarithm on both sides. Because, the base of the natural logarithm is e.
ln e^{x - 1} = ln 10
Use the power rule of logarithm.
(x - 1)ln e = ln 10
(x - 1)(1) = ln 10
x - 1 = ln 10
Add 1 to both sides.
x = ln 10 + 1
x = 2.30258.... + 1
x = 3.3026
Problem 5 :
16^{n - 7} + 5 = 24
Solution :
16^{n - 7} + 5 = 24
Subtract 5 from both sides.
16^{n - 7} = 19
Take logarithm with base 10 on both sides.
log 16^{n - 7} = log 19
Use the power rule of logarithm.
(n - 7)log 16 = log 19
Divide both sides by log 16.
x = 8.0620
Problem 6 :
2e^{x + 3} = 2960
Solution :
2e^{x + 3} = 2960
Divide both sides by 2.
e^{x + 3} = 1480
Since the base on the left side is e, take natural logarithm on both sides. Because, the base of the natural logarithm is e.
ln e^{x + 3} = ln 1480
Use the power rule of logarithm.
(x + 3)ln e = ln 1480
(x + 3)(1) = ln 1480
x + 3 = ln 1480
Subtract 3 from both sides.
x = ln 1480 - 3
x = 7.29979.... - 3
x = 4.2998
Problem 7 :
9^{n + 10} + 3 = 81
Solution :
9^{n + 10} + 3 = 81
Subtract 3 from both sides.
9^{n + 10} = 78
Take logarithm with base 10 on both sides.
log 9^{n + 10} = log 78
Use the power rule of logarithm.
(n + 10)log 9 = log 78
Divide both sides by log 9.
x = -8.0172
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