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Amy Ferguson Moncure

H Algebra 2/Trig · Unit 7

Evaluate Logarithms

Find the exact value by hand. If the argument is 0 or negative, the log has no solution.

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Key Ideas Easy Medium Hard

logb a = c means bc = a

logb 1 = 0 and logb b = 1

The argument must be positive.

What to Remember

The answer is an exponent

log2 8 = 3 because 23 = 8.

Write the argument as a power of the base, then read the exponent.

Ones and the base itself

logb 1 = 0 because b0 = 1.

logb b = 1 because b1 = b.

ln e = 1 because e1 = e.

Fractions

A fraction or a negative result means a root or a negative exponent.

square root of 4 = 2, and (square root of 4)3 = 8, so log4 8 = 32.

Not defined

You cannot take a log of 0 or of a negative number.

Those expressions have no solution.

Level 1

Easy

Powers you can name

Example
Evaluate log2 64.

logb a = c means bc = a.

log2 64 asks: 2 to what power equals 64?

2⁶ = 64

The power is 6.

6

Answer: 6

1
Evaluate log5 625.
Show Answer

logb a = c means bc = a.

log5 625 asks: 5 to what power equals 625?

5⁴ = 625

The power is 4.

4

Answer: 4

2
Evaluate log3 243.
Show Answer

logb a = c means bc = a.

log3 243 asks: 3 to what power equals 243?

3⁵ = 243

The power is 5.

5

Answer: 5

3
Evaluate log7 1.
Show Answer

logb a = c means bc = a.

log7 1 asks: 7 to what power equals 1?

7⁰ = 1

The power is 0.

0

Answer: 0

4
Evaluate log4 4.
Show Answer

logb a = c means bc = a.

log4 4 asks: 4 to what power equals 4?

4¹ = 4

The power is 1.

1

Answer: 1

5
Evaluate ln e.
Show Answer

ln e means loge e.

e1 = e.

The power is 1.

Answer: 1

Level 2

Medium

Fractions and roots

Example
Evaluate log4 8.

logb a = c means bc = a.

log4 8 asks: 4 to what power equals 8?

square root of 4 = 2

(square root of 4)3 = 2³ = 8

So 4 to the power 32 equals 8.

32

Answer: 32

1
Evaluate log9 27.
Show Answer

logb a = c means bc = a.

log9 27 asks: 9 to what power equals 27?

square root of 9 = 3

(square root of 9)3 = 3³ = 27

So 9 to the power 32 equals 27.

32

Answer: 32

2
Evaluate log8 16.
Show Answer

logb a = c means bc = a.

log8 16 asks: 8 to what power equals 16?

2³ = 8, so the cube root of 8 is 2.

24 = 16

Four of those cube-root steps make 16, so the power is 43.

43

Answer: 43

3
Evaluate log2 (132).
Show Answer

logb a = c means bc = a.

log2 (132) asks: 2 to what power equals 132?

2⁵ = 32

2−5 = 132

The power is −5.

−5

Answer: −5

4
Evaluate log5 (1125).
Show Answer

logb a = c means bc = a.

log5 (1125) asks: 5 to what power equals 1125?

5³ = 125

5−3 = 1125

The power is −3.

−3

Answer: −3

5
Evaluate log16 8.
Show Answer

logb a = c means bc = a.

log16 8 asks: 16 to what power equals 8?

16 = 2⁴ and 8 = 2³

4 · 34 = 3, so 16 to the power 34 equals 8.

34

Answer: 34

Level 3

Hard

If possible

Example
Evaluate log3 (−27), if possible.

A logarithm is defined only when the argument is positive.

−27 is negative, so it is not a legal argument.

No solution

Answer: No solution

1
Evaluate log4 (−16), if possible.
Show Answer

A logarithm is defined only when the argument is positive.

−16 is negative.

No solution

Answer: No solution

2
Evaluate log5 (0), if possible.
Show Answer

A logarithm is defined only when the argument is positive.

The argument is 0.

No solution

Answer: No solution

3
Evaluate log(−25), if possible.
Show Answer

A logarithm is defined only when the argument is positive.

−25 is negative. A common log still needs a positive argument.

No solution

Answer: No solution

4
Evaluate ln(0), if possible.
Show Answer

A logarithm is defined only when the argument is positive.

The argument is 0.

No solution

Answer: No solution

5
Evaluate log9 (127).
Show Answer

logb a = c means bc = a.

log9 (127) asks: 9 to what power equals 127?

(square root of 9)3 = 27

So 9 to the power 32 equals 27.

A negative exponent takes the reciprocal.

9 to the power −32 equals 127.

The power is −32.

−32

Answer: −32

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