Change of base formula worksheet for Precalculus

Precalculus uses change of base two ways: to get decimal values of logs in any base, to three decimal places, and to rewrite a log in any base as a quotient of common logs. It pairs with the log properties, as in writing log 144 in terms of log 2 and log 3.

Also taught in Algebra 2, at its own level and with its own examples.

Select your difficulty

Pick more than one for a sheet that mixes them.

The preview can look cramped on a phone. The PDF and printed sheet come out normal.



Sigma Prep · sigmaprep.io/worksheets Name Change of Base Date Period Rewrite or evaluate each expression. Rewrite with common logarithms Rewrite using common logarithms: $\log_{11} (663)$ Rewrite using common logarithms: $\log_{12} (401)$ Rewrite using common logarithms: $\log_{8} (1099)$ Rewrite using common logarithms: $\log_{12} (557)$ Rewrite using common logarithms: $\log_{8} (300)$ Rewrite using common logarithms: $\log_{12} (1301)$ Rewrite using common logarithms: $\log_{11} (1319)$ Write in terms of $a$ and $b$ If $\log 3 = a$ and $\log 5 = b$ , write $\log 3375$ in terms of $a$ and $b$ . If $\log 2 = a$ and $\log 7 = b$ , write $\log \frac{16}{7}$ in terms of $a$ and $b$ . If $\log 2 = a$ and $\log 3 = b$ , write $\log 144$ in terms of $a$ and $b$ . If $\log 5 = a$ and $\log 7 = b$ , write $\log \frac{625}{49}$ in terms of $a$ and $b$ . If $\log 3 = a$ and $\log 7 = b$ , write $\log 9$ in terms of $a$ and $b$ . If $\log 2 = a$ and $\log 11 = b$ , write $\log \frac{4}{11}$ in terms of $a$ and $b$ . If $\log 3 = a$ and $\log 7 = b$ , write $\log \frac{3}{49}$ in terms of $a$ and $b$ . Evaluate with change of base Use a calculator: $\log_{5} (323)$ to three decimal places Use a calculator: $\log_{7} (21)$ to three decimal places Use a calculator: $\log_{8} (201)$ to three decimal places Use a calculator: $\log_{6} (437)$ to three decimal places Use a calculator: $\log_{3} (821)$ to three decimal places Use a calculator: $\log_{2} (574)$ to three decimal places Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Change of Base

Date Period

Rewrite or evaluate each expression.

Rewrite with common logarithms
Rewrite using common logarithms: $\log_{11} (663)$
Rewrite using common logarithms: $\log_{12} (401)$
Rewrite using common logarithms: $\log_{8} (1099)$
Rewrite using common logarithms: $\log_{12} (557)$
Rewrite using common logarithms: $\log_{8} (300)$
Rewrite using common logarithms: $\log_{12} (1301)$
Rewrite using common logarithms: $\log_{11} (1319)$

Write in terms of $a$ and $b$
If $\log 3 = a$ and $\log 5 = b$ , write $\log 3375$ in terms of $a$ and $b$ .
If $\log 2 = a$ and $\log 7 = b$ , write $\log \frac{16}{7}$ in terms of $a$ and $b$ .
If $\log 2 = a$ and $\log 3 = b$ , write $\log 144$ in terms of $a$ and $b$ .
If $\log 5 = a$ and $\log 7 = b$ , write $\log \frac{625}{49}$ in terms of $a$ and $b$ .
If $\log 3 = a$ and $\log 7 = b$ , write $\log 9$ in terms of $a$ and $b$ .
If $\log 2 = a$ and $\log 11 = b$ , write $\log \frac{4}{11}$ in terms of $a$ and $b$ .
If $\log 3 = a$ and $\log 7 = b$ , write $\log \frac{3}{49}$ in terms of $a$ and $b$ .

Evaluate with change of base
Use a calculator: $\log_{5} (323)$ to three decimal places
Use a calculator: $\log_{7} (21)$ to three decimal places
Use a calculator: $\log_{8} (201)$ to three decimal places
Use a calculator: $\log_{6} (437)$ to three decimal places
Use a calculator: $\log_{3} (821)$ to three decimal places
Use a calculator: $\log_{2} (574)$ to three decimal places



Sigma Prep · sigmaprep.io/worksheets Name Change of BaseAnswer keyVersion 2 Date Period Rewrite or evaluate each expression. Rewrite with common logarithms Rewrite using common logarithms: $\log_{11} (663)$ $\frac{\log 663}{\log 11}$ Rewrite using common logarithms: $\log_{12} (401)$ $\frac{\log 401}{\log 12}$ Rewrite using common logarithms: $\log_{8} (1099)$ $\frac{\log 1099}{\log 8}$ Rewrite using common logarithms: $\log_{12} (557)$ $\frac{\log 557}{\log 12}$ Rewrite using common logarithms: $\log_{8} (300)$ $\frac{\log 300}{\log 8}$ Rewrite using common logarithms: $\log_{12} (1301)$ $\frac{\log 1301}{\log 12}$ Rewrite using common logarithms: $\log_{11} (1319)$ $\frac{\log 1319}{\log 11}$ Write in terms of $a$ and $b$ If $\log 3 = a$ and $\log 5 = b$ , write $\log 3375$ in terms of $a$ and $b$ . $3 a + 3 b$ If $\log 2 = a$ and $\log 7 = b$ , write $\log \frac{16}{7}$ in terms of $a$ and $b$ . $4 a - b$ If $\log 2 = a$ and $\log 3 = b$ , write $\log 144$ in terms of $a$ and $b$ . $4 a + 2 b$ If $\log 5 = a$ and $\log 7 = b$ , write $\log \frac{625}{49}$ in terms of $a$ and $b$ . $4 a - 2 b$ If $\log 3 = a$ and $\log 7 = b$ , write $\log 9$ in terms of $a$ and $b$ . $2 a$ If $\log 2 = a$ and $\log 11 = b$ , write $\log \frac{4}{11}$ in terms of $a$ and $b$ . $2 a - b$ If $\log 3 = a$ and $\log 7 = b$ , write $\log \frac{3}{49}$ in terms of $a$ and $b$ . $a - 2 b$ Evaluate with change of base Use a calculator: $\log_{5} (323)$ to three decimal places $3.590$ Use a calculator: $\log_{7} (21)$ to three decimal places $1.565$ Use a calculator: $\log_{8} (201)$ to three decimal places $2.550$ Use a calculator: $\log_{6} (437)$ to three decimal places $3.393$ Use a calculator: $\log_{3} (821)$ to three decimal places $6.108$ Use a calculator: $\log_{2} (574)$ to three decimal places $9.165$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Change of BaseAnswer keyVersion 2

Date Period

Rewrite or evaluate each expression.

Rewrite with common logarithms
Rewrite using common logarithms: $\log_{11} (663)$ $\frac{\log 663}{\log 11}$
Rewrite using common logarithms: $\log_{12} (401)$ $\frac{\log 401}{\log 12}$
Rewrite using common logarithms: $\log_{8} (1099)$ $\frac{\log 1099}{\log 8}$
Rewrite using common logarithms: $\log_{12} (557)$ $\frac{\log 557}{\log 12}$
Rewrite using common logarithms: $\log_{8} (300)$ $\frac{\log 300}{\log 8}$
Rewrite using common logarithms: $\log_{12} (1301)$ $\frac{\log 1301}{\log 12}$
Rewrite using common logarithms: $\log_{11} (1319)$ $\frac{\log 1319}{\log 11}$

Write in terms of $a$ and $b$
If $\log 3 = a$ and $\log 5 = b$ , write $\log 3375$ in terms of $a$ and $b$ . $3 a + 3 b$
If $\log 2 = a$ and $\log 7 = b$ , write $\log \frac{16}{7}$ in terms of $a$ and $b$ . $4 a - b$
If $\log 2 = a$ and $\log 3 = b$ , write $\log 144$ in terms of $a$ and $b$ . $4 a + 2 b$
If $\log 5 = a$ and $\log 7 = b$ , write $\log \frac{625}{49}$ in terms of $a$ and $b$ . $4 a - 2 b$
If $\log 3 = a$ and $\log 7 = b$ , write $\log 9$ in terms of $a$ and $b$ . $2 a$
If $\log 2 = a$ and $\log 11 = b$ , write $\log \frac{4}{11}$ in terms of $a$ and $b$ . $2 a - b$
If $\log 3 = a$ and $\log 7 = b$ , write $\log \frac{3}{49}$ in terms of $a$ and $b$ . $a - 2 b$

Evaluate with change of base
Use a calculator: $\log_{5} (323)$ to three decimal places $3.590$
Use a calculator: $\log_{7} (21)$ to three decimal places $1.565$
Use a calculator: $\log_{8} (201)$ to three decimal places $2.550$
Use a calculator: $\log_{6} (437)$ to three decimal places $3.393$
Use a calculator: $\log_{3} (821)$ to three decimal places $6.108$
Use a calculator: $\log_{2} (574)$ to three decimal places $9.165$

How to do these

Use a calculator: $\log_{3} (50)$ to three decimal places

Change of base: log 50 over log 3.
That is about 1.69897 ÷ 0.47712 = 3.5609...

The answer is $3.561$ .

Where students go wrong

Rounding the two logs before dividing. Using 1.70 and 0.48 gives 3.542, not 3.561; keep the full calculator values until the final division.

Also called change of base, log base conversion or expressing logs in terms of others.

What you can put on this worksheet

Rewrite with common logarithms: Rewrite using common logarithms: $\log_{3} (21)$ → $\frac{\log 21}{\log 3}$
Write one logarithm in terms of others: If $\log 3 = a$ and $\log 7 = b$ , write $\log 63$ in terms of $a$ and $b$ . → $2 a + b$
With a calculator: Use a calculator: $\log_{6} (29)$ to three decimal places → $1.879$
Decimals, ln and log directly, show two logs are equal: Approximate to four decimal places: $\log 1.7$ → $0.2304$

Questions about these worksheets

Yes. Every change of base formula sheet is free to print and free to download as a PDF. There is no account to make, no email to hand over and no limit on how many you take.

Yes. The answer key prints on a second page, and you can choose whether it shows each question beside its answer or just the answers in a list.

Yes. Print straight from the page, or download the PDF and print that. The sheet is laid out for paper rather than squeezed off a screen, so there is room to work under each question, and the answer key comes out on its own page.

Precalculus is usually taken around 11th or 12th grade, though schools vary and plenty of students meet it earlier or later. Pick the difficulty rather than the grade: the easy level suits a first lesson on it, the hard level suits review before a test.

Yes, as many as you like. The questions are built fresh each time rather than picked from a fixed set of files, so pressing Generate gives a new sheet. That is what you want for a second class, a retake, or two students sitting next to each other.

That is the closest comparison, and Kuta Software is good software. It is also paid software you install on a computer. These worksheets run in a browser tab for free, with no account and nothing to install. Like Kuta, the change of base formula questions are generated when you ask for them rather than pulled from a fixed set of files, so two classes never get the same sheet, and the answer key comes with it.

Yes. There are three levels, and you can tick more than one for a sheet that mixes them, in which case the questions come out easiest first. The harder levels are not just larger numbers: they ask for something the easy ones do not.

You pick from 4: rewrite with common logarithms, write one logarithm in terms of others, with a calculator and decimals, ln and log directly, show two logs are equal. Tick as many as you want and set how many of each, or let it spread them evenly.

Putting the base on top. It goes underneath, always. A quick check: log base 7 of 30 should be a bit under 2, and log 7 over log 30 is about 0.57, which is nowhere near.

Change of base formula worksheet for Precalculus

Select your difficulty

Rewrite with common logarithms

Write in terms of aaa and bbb

Evaluate with change of base

Rewrite with common logarithms

Write in terms of aaa and bbb

Evaluate with change of base

How to do these

Where students go wrong

What you can put on this worksheet

Questions about these worksheets

Are these change of base formula worksheets free?+

Do the change of base formula worksheets come with an answer key?+

Are these change of base formula worksheets printable?+

What grade are these change of base formula worksheets for?+

Can I get a different version of the same change of base formula worksheet?+

Is this a free alternative to Kuta Software for change of base formula?+

Can I choose how hard the change of base formula questions are?+

What kinds of question are on a change of base formula worksheet?+

What do students get wrong with change of base formula?+

Write in terms of $a$ and $b$

Write in terms of $a$ and $b$