Change of base formula worksheet

A calculator only has two log buttons, so any other base has to be rewritten as a quotient of those. The same trick lets you build one logarithm out of others you already know.

Select your difficulty

Pick more than one for a sheet that mixes them.



Sigma Prep · sigmaprep.io/worksheets Name Change of Base Date Period Rewrite each expression. Rewrite using common logarithms: $\log_{6} (40)$ If $\log 2 = a$ and $\log 7 = b$ , write $\log \frac{8}{49}$ in terms of $a$ and $b$ Rewrite using common logarithms: $\log_{6} (184)$ If $\log 3 = a$ and $\log 5 = b$ , write $\log 135$ in terms of $a$ and $b$ Rewrite using common logarithms: $\log_{4} (121)$ If $\log 2 = a$ and $\log 5 = b$ , write $\log \frac{2}{5}$ in terms of $a$ and $b$ Rewrite using common logarithms: $\log_{6} (194)$ If $\log 2 = a$ and $\log 5 = b$ , write $\log 2$ in terms of $a$ and $b$ Rewrite using common logarithms: $\log_{6} (200)$ If $\log 2 = a$ and $\log 5 = b$ , write $\log 10$ in terms of $a$ and $b$ Rewrite using common logarithms: $\log_{7} (78)$ If $\log 3 = a$ and $\log 7 = b$ , write $\log 441$ in terms of $a$ and $b$ Rewrite using common logarithms: $\log_{4} (192)$ If $\log 2 = a$ and $\log 3 = b$ , write $\log \frac{2}{3}$ in terms of $a$ and $b$ Rewrite using common logarithms: $\log_{4} (185)$ If $\log 2 = a$ and $\log 7 = b$ , write $\log 2$ in terms of $a$ and $b$ Rewrite using common logarithms: $\log_{6} (214)$ If $\log 2 = a$ and $\log 5 = b$ , write $\log 200$ in terms of $a$ and $b$ Rewrite using common logarithms: $\log_{4} (176)$ If $\log 5 = a$ and $\log 7 = b$ , write $\log 6125$ in terms of $a$ and $b$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Change of Base

Date Period

Rewrite each expression.

Rewrite using common logarithms: $\log_{6} (40)$
If $\log 2 = a$ and $\log 7 = b$ , write $\log \frac{8}{49}$ in terms of $a$ and $b$
Rewrite using common logarithms: $\log_{6} (184)$
If $\log 3 = a$ and $\log 5 = b$ , write $\log 135$ in terms of $a$ and $b$
Rewrite using common logarithms: $\log_{4} (121)$
If $\log 2 = a$ and $\log 5 = b$ , write $\log \frac{2}{5}$ in terms of $a$ and $b$
Rewrite using common logarithms: $\log_{6} (194)$
If $\log 2 = a$ and $\log 5 = b$ , write $\log 2$ in terms of $a$ and $b$
Rewrite using common logarithms: $\log_{6} (200)$
If $\log 2 = a$ and $\log 5 = b$ , write $\log 10$ in terms of $a$ and $b$
Rewrite using common logarithms: $\log_{7} (78)$
If $\log 3 = a$ and $\log 7 = b$ , write $\log 441$ in terms of $a$ and $b$
Rewrite using common logarithms: $\log_{4} (192)$
If $\log 2 = a$ and $\log 3 = b$ , write $\log \frac{2}{3}$ in terms of $a$ and $b$
Rewrite using common logarithms: $\log_{4} (185)$
If $\log 2 = a$ and $\log 7 = b$ , write $\log 2$ in terms of $a$ and $b$
Rewrite using common logarithms: $\log_{6} (214)$
If $\log 2 = a$ and $\log 5 = b$ , write $\log 200$ in terms of $a$ and $b$
Rewrite using common logarithms: $\log_{4} (176)$
If $\log 5 = a$ and $\log 7 = b$ , write $\log 6125$ in terms of $a$ and $b$



Sigma Prep · sigmaprep.io/worksheets Name Change of BaseAnswer keyVersion 1 Date Period Rewrite each expression. Rewrite using common logarithms: $\log_{6} (40)$ $\frac{\log 40}{\log 6}$ If $\log 2 = a$ and $\log 7 = b$ , write $\log \frac{8}{49}$ in terms of $a$ and $b$ $3 a - 2 b$ Rewrite using common logarithms: $\log_{6} (184)$ $\frac{\log 184}{\log 6}$ If $\log 3 = a$ and $\log 5 = b$ , write $\log 135$ in terms of $a$ and $b$ $3 a + b$ Rewrite using common logarithms: $\log_{4} (121)$ $\frac{\log 121}{\log 4}$ If $\log 2 = a$ and $\log 5 = b$ , write $\log \frac{2}{5}$ in terms of $a$ and $b$ $a - b$ Rewrite using common logarithms: $\log_{6} (194)$ $\frac{\log 194}{\log 6}$ If $\log 2 = a$ and $\log 5 = b$ , write $\log 2$ in terms of $a$ and $b$ $a$ Rewrite using common logarithms: $\log_{6} (200)$ $\frac{\log 200}{\log 6}$ If $\log 2 = a$ and $\log 5 = b$ , write $\log 10$ in terms of $a$ and $b$ $a + b$ Rewrite using common logarithms: $\log_{7} (78)$ $\frac{\log 78}{\log 7}$ If $\log 3 = a$ and $\log 7 = b$ , write $\log 441$ in terms of $a$ and $b$ $2 a + 2 b$ Rewrite using common logarithms: $\log_{4} (192)$ $\frac{\log 192}{\log 4}$ If $\log 2 = a$ and $\log 3 = b$ , write $\log \frac{2}{3}$ in terms of $a$ and $b$ $a - b$ Rewrite using common logarithms: $\log_{4} (185)$ $\frac{\log 185}{\log 4}$ If $\log 2 = a$ and $\log 7 = b$ , write $\log 2$ in terms of $a$ and $b$ $a$ Rewrite using common logarithms: $\log_{6} (214)$ $\frac{\log 214}{\log 6}$ If $\log 2 = a$ and $\log 5 = b$ , write $\log 200$ in terms of $a$ and $b$ $3 a + 2 b$ Rewrite using common logarithms: $\log_{4} (176)$ $\frac{\log 176}{\log 4}$ If $\log 5 = a$ and $\log 7 = b$ , write $\log 6125$ in terms of $a$ and $b$ $3 a + 2 b$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Change of BaseAnswer keyVersion 1

Date Period

Rewrite each expression.

Rewrite using common logarithms: $\log_{6} (40)$ $\frac{\log 40}{\log 6}$
If $\log 2 = a$ and $\log 7 = b$ , write $\log \frac{8}{49}$ in terms of $a$ and $b$ $3 a - 2 b$
Rewrite using common logarithms: $\log_{6} (184)$ $\frac{\log 184}{\log 6}$
If $\log 3 = a$ and $\log 5 = b$ , write $\log 135$ in terms of $a$ and $b$ $3 a + b$
Rewrite using common logarithms: $\log_{4} (121)$ $\frac{\log 121}{\log 4}$
If $\log 2 = a$ and $\log 5 = b$ , write $\log \frac{2}{5}$ in terms of $a$ and $b$ $a - b$
Rewrite using common logarithms: $\log_{6} (194)$ $\frac{\log 194}{\log 6}$
If $\log 2 = a$ and $\log 5 = b$ , write $\log 2$ in terms of $a$ and $b$ $a$
Rewrite using common logarithms: $\log_{6} (200)$ $\frac{\log 200}{\log 6}$
If $\log 2 = a$ and $\log 5 = b$ , write $\log 10$ in terms of $a$ and $b$ $a + b$
Rewrite using common logarithms: $\log_{7} (78)$ $\frac{\log 78}{\log 7}$
If $\log 3 = a$ and $\log 7 = b$ , write $\log 441$ in terms of $a$ and $b$ $2 a + 2 b$
Rewrite using common logarithms: $\log_{4} (192)$ $\frac{\log 192}{\log 4}$
If $\log 2 = a$ and $\log 3 = b$ , write $\log \frac{2}{3}$ in terms of $a$ and $b$ $a - b$
Rewrite using common logarithms: $\log_{4} (185)$ $\frac{\log 185}{\log 4}$
If $\log 2 = a$ and $\log 7 = b$ , write $\log 2$ in terms of $a$ and $b$ $a$
Rewrite using common logarithms: $\log_{6} (214)$ $\frac{\log 214}{\log 6}$
If $\log 2 = a$ and $\log 5 = b$ , write $\log 200$ in terms of $a$ and $b$ $3 a + 2 b$
Rewrite using common logarithms: $\log_{4} (176)$ $\frac{\log 176}{\log 4}$
If $\log 5 = a$ and $\log 7 = b$ , write $\log 6125$ in terms of $a$ and $b$ $3 a + 2 b$

How to do these

Rewrite using common logarithms: $\log_{7} (30)$

Change of base: the log of the inside over the log of the base.

The answer is $\frac{\log 30}{\log 7}$ .

Where students go wrong

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.

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 $\log 2 = a$ and $\log 7 = b$ , write $\log 28$ in terms of $a$ and $b$ → $2 a + b$

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.

Algebra 2 is usually taken around 10th or 11th 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 two 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 2: rewrite with common logarithms and write one logarithm in terms of others. 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

Select your difficulty

Change of Base

Change of BaseAnswer keyVersion 1

How to do these

Where students go wrong

What you can put on this worksheet

Questions about these worksheets

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