Solve log₇5 - log₇2: Base 7 Logarithm Subtraction

Q: Why can't I just subtract 5 - 2 = 3 and write log₇3?

Logarithms don't work like regular numbers! When you subtract logarithms with the same base, you're actually dividing the numbers inside. So $ \log_7 5 - \log_7 2 = \log_7 \left(\frac{5}{2}\right) $, not log₇3.

Q: What if the bases were different?

You cannot use the quotient rule when bases are different! For example, log₇5 - log₂2 cannot be simplified this way. The quotient rule only works with identical bases.

Q: How do I remember when to divide vs multiply inside the log?

Think of it this way: Subtraction of logs = Division inside. Addition of logs = Multiplication inside. The operations are opposite!

Q: Do I need to convert 2.5 to a fraction?

Both $ \log_7 \frac{5}{2} $ and $ \log_7 2.5 $ are correct! Since $ \frac{5}{2} = 2.5 $, they're the same value. Use whichever form matches your answer choices.

Q: Can I check this without a calculator?

Yes! You can't easily find the numerical value of log₇2.5, but you can verify your algebra is correct. Check that $ \frac{5}{2} = 2.5 $ and that you applied the quotient rule properly.

Logarithm Properties with Quotient Rule

$\log_75-\log_72=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Solve

00:09 We will use the formula for subtracting logarithms

00:17 Subtracting logarithms equals the logarithm of the quotient of numbers

00:24 We will use this formula in our exercise

00:34 And this is the solution to the question

Step-by-step written solution

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Understand the problem

$\log_75-\log_72=$

Step-by-step solution

To solve the problem, let's use the rules of logarithms:

Step 1: Recognize that we are dealing with the subtraction of logarithms sharing the same base, which calls for the identity $\log_b M - \log_b N = \log_b \left(\frac{M}{N}\right)$ .
Step 2: Apply this identity to the expression $\log_7 5 - \log_7 2$ .
Step 3: Realize that this can thus be expressed as a single logarithm: $\log_7 \left(\frac{5}{2}\right)$ .
Step 4: Simplify the fraction, yielding $\log_7 2.5$ .

Therefore, the simplification results in the expression: $\log_7 2.5$ .

This matches the correct answer from the given choices.

Final Answer

$\log_72.5$

Key Points to Remember

Essential concepts to master this topic

Rule: Subtraction of same-base logarithms becomes division inside logarithm
Technique: $\log_7 5 - \log_7 2 = \log_7 \left(\frac{5}{2}\right)$
Check: Verify $\frac{5}{2} = 2.5$ matches the correct answer ✓

Common Mistakes

Avoid these frequent errors

Adding or multiplying the numbers inside instead of using quotient rule
Don't calculate log₇5 - log₇2 as log₇3 or log₇10! This ignores the logarithm properties completely and gives wrong results. Always use the quotient rule: log_b M - log_b N = log_b(M/N).

Practice Quiz

Test your knowledge with interactive questions

$\log_{10}3+\log_{10}4=$

FAQ

Everything you need to know about this question

Why can't I just subtract 5 - 2 = 3 and write log₇3?

Logarithms don't work like regular numbers! When you subtract logarithms with the same base, you're actually dividing the numbers inside. So $\log_7 5 - \log_7 2 = \log_7 \left(\frac{5}{2}\right)$ , not log₇3.

What if the bases were different?

You cannot use the quotient rule when bases are different! For example, log₇5 - log₂2 cannot be simplified this way. The quotient rule only works with identical bases.

How do I remember when to divide vs multiply inside the log?

Think of it this way: Subtraction of logs = Division inside. Addition of logs = Multiplication inside. The operations are opposite!

Do I need to convert 2.5 to a fraction?

Both $\log_7 \frac{5}{2}$ and $\log_7 2.5$ are correct! Since $\frac{5}{2} = 2.5$ , they're the same value. Use whichever form matches your answer choices.

Can I check this without a calculator?

Yes! You can't easily find the numerical value of log₇2.5, but you can verify your algebra is correct. Check that $\frac{5}{2} = 2.5$ and that you applied the quotient rule properly.

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Solve log₇5 - log₇2: Base 7 Logarithm Subtraction

Logarithm Properties with Quotient Rule

❤️ Continue Your Math Journey!

Step-by-step video solution

Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

Why can't I just subtract 5 - 2 = 3 and write log₇3?

What if the bases were different?

How do I remember when to divide vs multiply inside the log?

Do I need to convert 2.5 to a fraction?

Can I check this without a calculator?

🌟 Unlock Your Math Potential

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