1. use a property of logarithms to evaluate log 3 9 2 2. use log 5 ≈ 0.699 and log 6 ≈ 0.778 to...

1. Use a property of logarithms to evaluate log392

2. Use log 5 ≈ 0.699 and log 6 ≈ 0.778 to approximate the value of log 150.

3. Expand ln 732x4.

4. Condense 2log5y – 2log55 + ½log5x.

5. Use the change of base formula to evaluate log820.

1Algebra II

Solving Exponential and Logarithmic Equations

Algebra II

To solve equations you must undo what is being done

Exponential and logarithmic functions are inverse operations

You undo “exponentiating” by ”logarizing” and you undo “logarizing” by “exponentiating”

3Algebra II

1. ax = ay If x = y2. Logbx = logby if x = yExamples:

If 2x = 23 then x = 3 If ln x = 3 then e lnx = e3, so x = e3 If log3x = log3(5), then x = 5 If ex = 7 then ln ex = ln7 , so x = ln7

4Algebra II

Algebra II 5

Algebra II 6

83 xx

82 x4x

1432 xx

2x

Algebra II 7

xx 314 1x12 x

8Algebra II

9Algebra II

2x

42 x

521 log01 log 2 x

152 log2 x2 2

152102 x

10Algebra II

1xe

0x

44 xe21ln ln 2 xe

12ln 2 x2 2

122 xe

11Algebra II

133 xx123 x

x22 1x

12Algebra II

121 xx11 x

2xNo Solution

(Extraneous solution)

13Algebra II

1144 xx1154 x

x515 3x

14Algebra II

9x

15Algebra II

23x

22 log5 x

16Algebra II

4

52 ln x

17Algebra II

Check both

solutions!

18Algebra II

19Algebra II

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21Algebra II

22Algebra II

1. ln x = x2 – 2 2. Log3x = x2 – 2

23Algebra II

The population size y of a community of lemmings varies according to the relationship y = y0e0.15t. In this formula, t

is time in months and y0 is the initial population at the time 0. Estimate the population after 8 months if there

were originally 2000 lemmings.

y = y0e0.15t

y = 2000e0.15(8)

y = 2000e1.2

y ≈ 6640.2339

In 8 months, the population will be approximately 6640 lemmings.

Algebra II 24

How long does it take an investment of $5,000 to double if it is invested at 4%, compounded

quarterly?P = $5000r = 4% or 0.04Compounded quarterly = 4 times per year, n = 4.The investment doubles, so A must be $10,000.

Substitute these values and solve for t.

Algebra II 25

It takes more than17 years for themoney to double

in value.

Algebra II 26

27Algebra II

Solve each equation. Be sure to check solutions!!

1.32x = 27x+2

2.5e3x + 2 = 17

3.log4(5x – 11) = log4(3 – 2x)

4.-3 ln = 4