HONORS ALGEBRA 2 : Problems on Solving Exponential Equations with Logarithms

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

Video Solution

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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