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Amy Ferguson Moncure

H Algebra 2/Trig · Unit 7

Exponential and Logarithmic Form

Rewrite an exponential equation as a log, or a log equation as an exponential. Stop when the form changes. Do not solve for a new variable.

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Key Ideas Easy Medium Hard

bc = a

logb a = c

ln means loge. log with no base means log10.

What to Remember

The two forms name the same fact

bc = a and logb a = c are the same statement.

The base stays the base. The log answer is the exponent.

Common log and natural log

log with no base means log10.

ln means loge. So ln m = t means et = m.

Level 1

Easy

Numbers only

Example
Write 4³ = 64 in logarithmic form.

bc = a means logb a = c.

4³ = 64

log4 64 = 3

Answer: log4 64 = 3

1
Write 2⁶ = 64 in logarithmic form.
Show Answer

bc = a means logb a = c.

2⁶ = 64

log2 64 = 6

Answer: log2 64 = 6

2
Write 6³ = 216 in logarithmic form.
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bc = a means logb a = c.

6³ = 216

log6 216 = 3

Answer: log6 216 = 3

3
Write log7 49 = 2 in exponential form.
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logb a = c means bc = a.

log7 49 = 2

7² = 49

Answer: 7² = 49

4
Write log 10,000 = 4 in exponential form.
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logb a = c means bc = a.

log 10,000 = 4

10⁴ = 10,000

Answer: 10⁴ = 10,000

5
Write 3⁵ = 243 in logarithmic form.
Show Answer

bc = a means logb a = c.

3⁵ = 243

log3 243 = 5

Answer: log3 243 = 5

Level 2

Medium

A variable, a common log, or a natural log

Example
Write et = p in logarithmic form.

bc = a means logb a = c.

et = p

ln p = t

Answer: ln p = t

1
Write ln w = 6 in exponential form.
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logb a = c means bc = a.

ln w = 6

e6 = w

Answer: e6 = w

2
Write log y = 3 in exponential form.
Show Answer

logb a = c means bc = a.

log y = 3

10³ = y

Answer: 10³ = y

3
Write 5n = p in logarithmic form.
Show Answer

bc = a means logb a = c.

5n = p

log5 p = n

Answer: log5 p = n

4
Write 9k = 4 in logarithmic form.
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bc = a means logb a = c.

9k = 4

log9 4 = k

Answer: log9 4 = k

5
Write log2 m = 5 in exponential form.
Show Answer

logb a = c means bc = a.

log2 m = 5

2⁵ = m

Answer: 2⁵ = m

Level 3

Hard

The exponent or the argument is an expression

Example
Write e2x = 7 in logarithmic form.

bc = a means logb a = c.

e2x = 7

ln 7 = 2x

Answer: ln 7 = 2x

1
Write log3 (x − 1) = 4 in exponential form.
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logb a = c means bc = a.

log3 (x − 1) = 4

3⁴ = x − 1

Answer: 3⁴ = x − 1

2
Write 5x + 2 = y in logarithmic form.
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bc = a means logb a = c.

5x + 2 = y

log5 y = x + 2

Answer: log5 y = x + 2

3
Write ln(2x) = −3 in exponential form.
Show Answer

logb a = c means bc = a.

ln(2x) = −3

e−3 = 2x

Answer: e−3 = 2x

4
Write log6 (x2) = 4 in exponential form.
Show Answer

logb a = c means bc = a.

log6 (x2) = 4

6⁴ = x2

Answer: 6⁴ = x2

5
Write log2 (x + 5) = y in exponential form.
Show Answer

logb a = c means bc = a.

log2 (x + 5) = y

2y = x + 5

Answer: 2y = x + 5

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