Solve log₅3 - log₅2: Working with Base-5 Logarithm Subtraction

Logarithm Subtraction with Property Application

log⁡53−log⁡52= \log_53-\log_52=

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations
00:00 Solve
00:04 We will use the formula for subtracting logarithms
00:13 Subtracting logarithms equals the logarithm of the quotient of numbers
00:19 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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1

Understand the problem

log⁡53−log⁡52= \log_53-\log_52=

2

Step-by-step solution

To solve the problem, we employ the property of logarithms for subtraction:

  • Step 1: Recognize the expression log⁡53−log⁡52 \log_5 3 - \log_5 2 .
  • Step 2: Apply the logarithmic property for subtraction, log⁡ba−log⁡bc=log⁡b(ac) \log_b a - \log_b c = \log_b \left( \frac{a}{c} \right) .
  • Step 3: Substitute into the property: log⁡53−log⁡52=log⁡5(32) \log_5 3 - \log_5 2 = \log_5 \left( \frac{3}{2} \right) .

By applying the property, we simplify the expression to log⁡532 \log_5 \frac{3}{2} . This is equivalent to log⁡51.5 \log_5 1.5 . Therefore:

Therefore, the result of the expression is log⁡51.5 \log_5 1.5 .

3

Final Answer

log⁡51.5 \log_51.5

Key Points to Remember

Essential concepts to master this topic
  • Property: log_b(a) - log_b(c) = log_b(a/c) for same bases
  • Technique: Apply subtraction rule: log₅3 - log₅2 = log₅(3/2)
  • Check: Verify 3/2 = 1.5, so log₅(3/2) = log₅1.5 ✓

Common Mistakes

Avoid these frequent errors
  • Converting to different bases or calculating decimal values
    Don't change log₅3 - log₅2 to log₁₀1.5 or try calculating decimal approximations = wrong answer! This ignores the base-5 requirement and creates unnecessary complexity. Always keep the same base and apply the subtraction property directly: log₅3 - log₅2 = log₅(3/2).

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 the numbers inside the logarithms?

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Logarithms don't work like regular subtraction! You can't do log₅3 - log₅2 = log₅(3-2) = log₅1. Instead, use the property: subtraction of logs equals the log of division.

How do I remember the subtraction property?

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Think: "Subtraction becomes Division" - When you subtract logs with the same base, it becomes the log of a fraction. log⁡ba−log⁡bc=log⁡b(ac) \log_b a - \log_b c = \log_b \left(\frac{a}{c}\right)

Is log₅(3/2) the same as log₅1.5?

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Yes! Since 32=1.5 \frac{3}{2} = 1.5 , both expressions are exactly the same. You can write the answer as either log₅(3/2) or log₅1.5.

What if the bases were different?

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If the bases are different (like log₅3 - log₂2), you cannot use the subtraction property directly. The logarithms must have the same base to combine them.

Can I calculate the exact decimal value?

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You could use a calculator, but it's not necessary! The exact answer log₅1.5 is perfectly acceptable and often preferred in mathematics because it shows the precise relationship.

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