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Rewriting Logarithms - Printable Mathematics Worksheets - Wayground

Rewriting Logarithms

20 questions

9th - 12th Grade

Mathematics

Rewriting Logarithms is a free printable mathematics worksheet for students in Grades 9–12. It provides 20 practice questions—19 multiple-choice and 1 multiple-select—focused on converting between logarithmic and exponential forms, identifying the base of a common logarithm, and interpreting whether an exponential relationship represents growth or decay. Students use the relationship between a logarithm’s base, argument, and exponent to rewrite equations such as log₂32 = 5 and 3⁴ = 81 in the alternate form. Items also include negative exponents, decimal values, variable notation, and selecting the correct transformation from plausible choices, helping students check the order of each component. The worksheet includes a complete answer key and is available as a free printable resource for class practice, independent work, review, or assessment.

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Rewriting Logarithms

Total questions: 20

Name
Class
Date
1.

Rewrite in exponential form. log⁡6 t=m\log_{6\ }t=m  

a)

mt=6m^t=6  

b)

tm=6t^m=6  

c)

6t=m6^t=m  

d)

6m=t6^m=t  

2.

Rewrite the equation into logarithmic form:

62 = 36

a)

log6 2 = 36

b)

log6 36 = 2

c)

ln 36 = 2

d)

636 = 2

3.
Write in exponential form.
log232 = 5
a)
2-5 = 32
b)
232 = 5
c)
25 = 32
d)
325 = 2
4.
Write log2 0.25 = -2 in exponential form
a)
-22  = 0.25
b)
-20.25  = 2
c)
2-2  = 0.25
d)
No correct answer
5.

Rewrite 34 = 81 in logarithmic form.

a)

log3 4 = 81

b)

log81 3 = 4

c)

log3 81 = 4

d)

log4 81 = 3

6.

Change to Exponential Form:

log6 36 = 2

a)

26=36

b)

62=36

c)

362=6

d)

366=2

7.

Write in logarithm form
6−3=12166^{-3}=\frac{1}{216}  

a)

log⁡6 −3=1216\log_6\ -3=\frac{1}{216}  

b)

log⁡−3 6=1216\log_{-3}\ 6=\frac{1}{216}  

c)

log⁡6 1216=−3\log_6\ \frac{1}{216}=-3  

d)

log⁡1216 6=−3\log_{\frac{1}{216}}\ 6=-3

8.

Rewrite in log form: 170=117^0=1  

a)

log⁡170=1\log_{17}0=1  

b)

log⁡171=0\log_{17}1=0  

c)

log⁡117=0\log_117=0  

d)

log⁡01=17\log_01=17  

9.

Rewrite in exponential form: log⁡4 164=−3\log_4\ \frac{1}{64}=-3  

a)

(164)−3=4\left(\frac{1}{64}\right)^{-3}=4  

b)

4164=−34^{\frac{1}{64}}=-3  

c)

(−3)4=164\left(-3\right)^4=\frac{1}{64}  

d)

4−3=1644^{-3}=\frac{1}{64}  

10.

what is the base for a log equation of the form log⁡ 1100=−2\log\ \frac{1}{100}=-2  

a)

10

b)

1

c)

e

d)

-1

11.
Rewrite logpt = m in exponential form.
a)
pt = m
b)
tm = p
c)
mt = p
d)
pm = t
12.
Is this exponential growth or decay?
Is this exponential growth or decay?
a)
Growth
b)
Decay
13.

Rewrite the equation in logarithmic form:
122=14412^2=144  

a)

log⁡122=144\log_{12}2=144  

b)

log⁡2144=12\log_2144=12  

c)

log⁡12144=2\log_{12}144=2  

d)

log⁡14412=2\log_{144}12=2  

14.

Rewrite the equation in logarithmic form:
9−2=1819^{-2}=\frac{1}{81}  

a)

log⁡29=181\log_29=\frac{1}{81}  

b)

log⁡9 (−2)=181\log_9\ \left(-2\right)=\frac{1}{81}  

c)

log⁡2 (181)=9\log_2\ \left(\frac{1}{81}\right)=9  

d)

log⁡9 181=−2\log_9\ \frac{1}{81}=-2  

15.

Rewrite the equation in logarithmic form:
8b=a8^b=a  

a)

log⁡a8=b\log_a8=b  

b)

log⁡8a=b\log_8a=b  

c)

log⁡ba=8\log_ba=8  

d)

log⁡8b=a\log_8b=a  

16.
Any number written to the 0 power is always 1
Example: 80=1
a)
True
b)
False
17.

Select the graph(s) that show an exponential function.

a)
Select the graph(s) that show an exponential function .
b)
Select the graph(s) that show an exponential function .
c)
Select the graph(s) that show an exponential function .
18.
Simplify the exponential expression.
Simplify the exponential expression.
a)
2x6y12
b)
2x5y7
c)
8x6y12
d)
8x5y7
19.
Is the graph linear, exponential or neither?
Is the graph linear, exponential or neither?
a)
Linear
b)
Exponential
c)
Neither
20.
What type of function is y = 7(5/4)x?
a)
Exponential Growth
b)
Exponential Decay
c)
Linear
d)
None of the above
WaygroundAnswer Key

Rewriting Logarithms

Total questions: 20

1.
d)

6m=t6^m=t  

2.
b)

log6 36 = 2

3.
c) 25 = 32
4.
c) 2-2  = 0.25
5.
c)

log3 81 = 4

6.
b)

62=36

7.
c)

log⁡6 1216=−3\log_6\ \frac{1}{216}=-3  

8.
b)

log⁡171=0\log_{17}1=0  

9.
d)

4−3=1644^{-3}=\frac{1}{64}  

10.
a)

10

11.
d) pm = t
12.
b) Decay
13.
c)

log⁡12144=2\log_{12}144=2  

14.
d)

log⁡9 181=−2\log_9\ \frac{1}{81}=-2  

15.
b)

log⁡8a=b\log_8a=b  

16.
a) True
17.
c)
18.
c) 8x6y12
19.
a) Linear
20.
a) Exponential Growth

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FAQs

What does this Rewriting Logarithms worksheet help students practice?

This worksheet helps Grades 9–12 students convert equations between logarithmic and exponential form by identifying the base, exponent, and resulting value in the correct positions. Its 20 questions include multiple-choice and multiple-select items with positive and negative exponents, decimals, and variables, plus an item distinguishing exponential growth from decay.

How can I use this worksheet to teach students to rewrite logarithms?

Introduce the structure log_b a = c ↔ b^c = a, then model one conversion in each direction before assigning the worksheet. Ask students to label the base, argument, and exponent in examples such as log₂32 = 5 or 3⁴ = 81, and have them explain why each selected option preserves those roles. Use the items with negative exponents and 0.25 to connect the form change to powers less than 1.

What mistakes should I watch for on this logarithms worksheet?

Watch for students reversing the base and exponent, confusing the logarithm’s argument with its result, or selecting an option that changes the order of the three values. Students may also mishandle negative exponents in examples such as log₂0.25 = −2 or overlook that a common logarithm has base 10. On the growth-or-decay item, check that students distinguish the requested classification rather than treating every exponential relationship as growth.

How do I assign and grade this Rewriting Logarithms worksheet?

Wayground provides this worksheet as a printable PDF and as a digital quiz, and it includes a complete answer key for the 20 items. Printable submissions can be graded by scanning student work with the Wayground for Teachers app, while the digital quiz supports assigning the questions online.

How can I differentiate this logarithms worksheet for my students?

In the worksheet’s Advanced Settings, adjust font spacing and font size to give students more room to compare the base, exponent, and result in each conversion item, and enable a dyslexia-friendly font if helpful. You can also translate the logarithm and exponential-form prompts into another language while keeping the mathematical notation intact.

Where can I find more free worksheets like this from Wayground?

Wayground offers a broad collection of free printable worksheets and practice problems across subjects, with downloadable PDFs and answer keys to support mastery of essential skills and concepts. Browse more resources at https://wayground.com/en-us/worksheets.

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