Writing logs in terms of others worksheet

Break the number into the two you were given, and the log rules do the rest: multiplying becomes adding, dividing becomes subtracting, and a power becomes a multiplier.

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Sigma Prep · sigmaprep.io/worksheets Name Writing Logs in Terms of Others Date Period Write each logarithm in terms of the two letters given. If $\log_{7} 2 = a$ and $\log_{7} 3 = c$ , write $\log_{7} 24$ . If $\log_{2} 5 = a$ and $\log_{2} 7 = c$ , write $\log_{2} \frac{1}{35}$ . If $\log_{7} 2 = a$ and $\log_{7} 3 = c$ , write $\log_{7} \frac{4}{3}$ . If $\log_{2} 3 = a$ and $\log_{2} 7 = c$ , write $\log_{2} 21$ . If $\log_{13} 3 = a$ and $\log_{13} 5 = c$ , write $\log_{13} \frac{9}{25}$ . If $\log_{13} 2 = a$ and $\log_{13} 3 = c$ , write $\log_{13} 12$ . If $\log_{3} 2 = a$ and $\log_{3} 5 = c$ , write $\log_{3} \frac{1}{50}$ . If $\log_{3} 5 = a$ and $\log_{3} 7 = c$ , write $\log_{3} 1715$ . If $\log_{5} 2 = a$ and $\log_{5} 7 = c$ , write $\log_{5} \frac{49}{4}$ . If $\log_{7} 3 = a$ and $\log_{7} 5 = c$ , write $\log_{7} 1125$ . If $\log_{11} 2 = a$ and $\log_{11} 7 = c$ , write $\log_{11} \frac{49}{2}$ . If $\log_{7} 2 = a$ and $\log_{7} 3 = c$ , write $\log_{7} \frac{8}{3}$ . If $\log_{7} 2 = a$ and $\log_{7} 3 = c$ , write $\log_{7} 216$ . If $\log_{3} 5 = a$ and $\log_{3} 7 = c$ , write $\log_{3} 6125$ . If $\log_{13} 2 = a$ and $\log_{13} 7 = c$ , write $\log_{13} \frac{7}{2}$ . If $\log_{11} 2 = a$ and $\log_{11} 7 = c$ , write $\log_{11} \frac{2}{7}$ . If $\log_{5} 2 = a$ and $\log_{5} 7 = c$ , write $\log_{5} \frac{8}{49}$ . If $\log_{2} 5 = a$ and $\log_{2} 7 = c$ , write $\log_{2} \frac{1}{1225}$ . If $\log_{3} 2 = a$ and $\log_{3} 5 = c$ , write $\log_{3} 50$ . If $\log_{11} 3 = a$ and $\log_{11} 7 = c$ , write $\log_{11} \frac{1}{63}$ . Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Writing Logs in Terms of Others

Date Period

Write each logarithm in terms of the two letters given.

If $\log_{7} 2 = a$ and $\log_{7} 3 = c$ , write $\log_{7} 24$ .
If $\log_{2} 5 = a$ and $\log_{2} 7 = c$ , write $\log_{2} \frac{1}{35}$ .
If $\log_{7} 2 = a$ and $\log_{7} 3 = c$ , write $\log_{7} \frac{4}{3}$ .
If $\log_{2} 3 = a$ and $\log_{2} 7 = c$ , write $\log_{2} 21$ .
If $\log_{13} 3 = a$ and $\log_{13} 5 = c$ , write $\log_{13} \frac{9}{25}$ .
If $\log_{13} 2 = a$ and $\log_{13} 3 = c$ , write $\log_{13} 12$ .
If $\log_{3} 2 = a$ and $\log_{3} 5 = c$ , write $\log_{3} \frac{1}{50}$ .
If $\log_{3} 5 = a$ and $\log_{3} 7 = c$ , write $\log_{3} 1715$ .
If $\log_{5} 2 = a$ and $\log_{5} 7 = c$ , write $\log_{5} \frac{49}{4}$ .
If $\log_{7} 3 = a$ and $\log_{7} 5 = c$ , write $\log_{7} 1125$ .
If $\log_{11} 2 = a$ and $\log_{11} 7 = c$ , write $\log_{11} \frac{49}{2}$ .
If $\log_{7} 2 = a$ and $\log_{7} 3 = c$ , write $\log_{7} \frac{8}{3}$ .
If $\log_{7} 2 = a$ and $\log_{7} 3 = c$ , write $\log_{7} 216$ .
If $\log_{3} 5 = a$ and $\log_{3} 7 = c$ , write $\log_{3} 6125$ .
If $\log_{13} 2 = a$ and $\log_{13} 7 = c$ , write $\log_{13} \frac{7}{2}$ .
If $\log_{11} 2 = a$ and $\log_{11} 7 = c$ , write $\log_{11} \frac{2}{7}$ .
If $\log_{5} 2 = a$ and $\log_{5} 7 = c$ , write $\log_{5} \frac{8}{49}$ .
If $\log_{2} 5 = a$ and $\log_{2} 7 = c$ , write $\log_{2} \frac{1}{1225}$ .
If $\log_{3} 2 = a$ and $\log_{3} 5 = c$ , write $\log_{3} 50$ .
If $\log_{11} 3 = a$ and $\log_{11} 7 = c$ , write $\log_{11} \frac{1}{63}$ .



Sigma Prep · sigmaprep.io/worksheets Name Writing Logs in Terms of OthersAnswer keyVersion 1 Date Period Write each logarithm in terms of the two letters given. If $\log_{7} 2 = a$ and $\log_{7} 3 = c$ , write $\log_{7} 24$ . $3 a + c$ If $\log_{2} 5 = a$ and $\log_{2} 7 = c$ , write $\log_{2} \frac{1}{35}$ . $- a - c$ If $\log_{7} 2 = a$ and $\log_{7} 3 = c$ , write $\log_{7} \frac{4}{3}$ . $2 a - c$ If $\log_{2} 3 = a$ and $\log_{2} 7 = c$ , write $\log_{2} 21$ . $a + c$ If $\log_{13} 3 = a$ and $\log_{13} 5 = c$ , write $\log_{13} \frac{9}{25}$ . $2 a - 2 c$ If $\log_{13} 2 = a$ and $\log_{13} 3 = c$ , write $\log_{13} 12$ . $2 a + c$ If $\log_{3} 2 = a$ and $\log_{3} 5 = c$ , write $\log_{3} \frac{1}{50}$ . $- a - 2 c$ If $\log_{3} 5 = a$ and $\log_{3} 7 = c$ , write $\log_{3} 1715$ . $a + 3 c$ If $\log_{5} 2 = a$ and $\log_{5} 7 = c$ , write $\log_{5} \frac{49}{4}$ . $- 2 a + 2 c$ If $\log_{7} 3 = a$ and $\log_{7} 5 = c$ , write $\log_{7} 1125$ . $2 a + 3 c$ If $\log_{11} 2 = a$ and $\log_{11} 7 = c$ , write $\log_{11} \frac{49}{2}$ . $- a + 2 c$ If $\log_{7} 2 = a$ and $\log_{7} 3 = c$ , write $\log_{7} \frac{8}{3}$ . $3 a - c$ If $\log_{7} 2 = a$ and $\log_{7} 3 = c$ , write $\log_{7} 216$ . $3 a + 3 c$ If $\log_{3} 5 = a$ and $\log_{3} 7 = c$ , write $\log_{3} 6125$ . $3 a + 2 c$ If $\log_{13} 2 = a$ and $\log_{13} 7 = c$ , write $\log_{13} \frac{7}{2}$ . $- a + c$ If $\log_{11} 2 = a$ and $\log_{11} 7 = c$ , write $\log_{11} \frac{2}{7}$ . $a - c$ If $\log_{5} 2 = a$ and $\log_{5} 7 = c$ , write $\log_{5} \frac{8}{49}$ . $3 a - 2 c$ If $\log_{2} 5 = a$ and $\log_{2} 7 = c$ , write $\log_{2} \frac{1}{1225}$ . $- 2 a - 2 c$ If $\log_{3} 2 = a$ and $\log_{3} 5 = c$ , write $\log_{3} 50$ . $a + 2 c$ If $\log_{11} 3 = a$ and $\log_{11} 7 = c$ , write $\log_{11} \frac{1}{63}$ . $- 2 a - c$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Writing Logs in Terms of OthersAnswer keyVersion 1

Date Period

Write each logarithm in terms of the two letters given.

If $\log_{7} 2 = a$ and $\log_{7} 3 = c$ , write $\log_{7} 24$ . $3 a + c$
If $\log_{2} 5 = a$ and $\log_{2} 7 = c$ , write $\log_{2} \frac{1}{35}$ . $- a - c$
If $\log_{7} 2 = a$ and $\log_{7} 3 = c$ , write $\log_{7} \frac{4}{3}$ . $2 a - c$
If $\log_{2} 3 = a$ and $\log_{2} 7 = c$ , write $\log_{2} 21$ . $a + c$
If $\log_{13} 3 = a$ and $\log_{13} 5 = c$ , write $\log_{13} \frac{9}{25}$ . $2 a - 2 c$
If $\log_{13} 2 = a$ and $\log_{13} 3 = c$ , write $\log_{13} 12$ . $2 a + c$
If $\log_{3} 2 = a$ and $\log_{3} 5 = c$ , write $\log_{3} \frac{1}{50}$ . $- a - 2 c$
If $\log_{3} 5 = a$ and $\log_{3} 7 = c$ , write $\log_{3} 1715$ . $a + 3 c$
If $\log_{5} 2 = a$ and $\log_{5} 7 = c$ , write $\log_{5} \frac{49}{4}$ . $- 2 a + 2 c$
If $\log_{7} 3 = a$ and $\log_{7} 5 = c$ , write $\log_{7} 1125$ . $2 a + 3 c$
If $\log_{11} 2 = a$ and $\log_{11} 7 = c$ , write $\log_{11} \frac{49}{2}$ . $- a + 2 c$
If $\log_{7} 2 = a$ and $\log_{7} 3 = c$ , write $\log_{7} \frac{8}{3}$ . $3 a - c$
If $\log_{7} 2 = a$ and $\log_{7} 3 = c$ , write $\log_{7} 216$ . $3 a + 3 c$
If $\log_{3} 5 = a$ and $\log_{3} 7 = c$ , write $\log_{3} 6125$ . $3 a + 2 c$
If $\log_{13} 2 = a$ and $\log_{13} 7 = c$ , write $\log_{13} \frac{7}{2}$ . $- a + c$
If $\log_{11} 2 = a$ and $\log_{11} 7 = c$ , write $\log_{11} \frac{2}{7}$ . $a - c$
If $\log_{5} 2 = a$ and $\log_{5} 7 = c$ , write $\log_{5} \frac{8}{49}$ . $3 a - 2 c$
If $\log_{2} 5 = a$ and $\log_{2} 7 = c$ , write $\log_{2} \frac{1}{1225}$ . $- 2 a - 2 c$
If $\log_{3} 2 = a$ and $\log_{3} 5 = c$ , write $\log_{3} 50$ . $a + 2 c$
If $\log_{11} 3 = a$ and $\log_{11} 7 = c$ , write $\log_{11} \frac{1}{63}$ . $- 2 a - c$

How to do these

If $\log_{5} 2 = a$ and $\log_{5} 3 = c$ , write $\log_{5} 12$

12 is 4 times 3, and 4 is 2 squared.
log of a product is a sum.
The square becomes a 2 in front.

The answer is $2 a + c$ .

Where students go wrong

Splitting a log of a sum. There is no rule for log(x + y): only products, quotients and powers come apart. If the number will not factor into the ones you were given, it cannot be done.

Also called log properties or expressing logarithms.

What you can put on this worksheet

One log in terms of others: If $\log_{7} 2 = a$ and $\log_{7} 5 = c$ , write $\log_{7} 100$ . → $2 a + 2 c$
Decimal givens, three letters, composite givens: Given $\log 12 = A, \log 8 = B$ . Find $\log ⁣ (\frac{3}{16})$ → $A - 2 B$

Questions about these worksheets

Yes. Every writing logs in terms of others 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 writing logs in terms of others 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 2: one log in terms of others and decimal givens, three letters, composite givens. Tick as many as you want and set how many of each, or let it spread them evenly.

Splitting a log of a sum. There is no rule for log(x + y): only products, quotients and powers come apart. If the number will not factor into the ones you were given, it cannot be done.

Writing logs in terms of others worksheet

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