Solve log₇4: Calculate the Base 7 Logarithm Value

Change of Base Formula with Natural Logarithms

log⁡74= \log_74=

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations
00:00 Solve
00:05 We will use the formula for logarithmic division
00:13 We get the log of the numerator with the denominator as the base
00:18 We will use this formula in our exercise
00:38 We will find the domain
00:55 We will use the formula to convert from log to ln
01:00 We will use this formula in our exercise
01:05 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution
1

Understand the problem

log⁡74= \log_74=

2

Step-by-step solution

To solve the problem of evaluating log⁡74\log_7 4, we will use the change-of-base formula for logarithms.

The change-of-base formula is:

  • log⁡ba=log⁡kalog⁡kb\log_b a = \frac{\log_k a}{\log_k b}, where kk can be any base, commonly chosen as 10 (common logs) or ee (natural logs).

We will choose natural logarithms (ln⁡\ln) for simplicity, therefore:

log⁡74=ln⁡4ln⁡7\log_7 4 = \frac{\ln 4}{\ln 7}

By applying the change-of-base formula, we find that the logarithm log⁡74\log_7 4 can be expressed as ln⁡4ln⁡7\frac{\ln 4}{\ln 7}.

Upon examining the provided choices, we identify that choice 2: ln⁡4ln⁡7\frac{\ln 4}{\ln 7} matches our result.

Therefore, the solution to the problem is ln⁡4ln⁡7\frac{\ln 4}{\ln 7}.

3

Final Answer

ln⁡4ln⁡7 \frac{\ln4}{\ln7}

Key Points to Remember

Essential concepts to master this topic
  • Formula: Use log⁡ba=ln⁡aln⁡b \log_b a = \frac{\ln a}{\ln b} to change any base
  • Technique: Convert log⁡74 \log_7 4 to ln⁡4ln⁡7 \frac{\ln 4}{\ln 7} using natural logs
  • Check: Verify 7ln⁡4ln⁡7=4 7^{\frac{\ln 4}{\ln 7}} = 4 using calculator ✓

Common Mistakes

Avoid these frequent errors
  • Placing numerator and denominator in wrong order
    Don't write ln⁡7ln⁡4 \frac{\ln 7}{\ln 4} for log⁡74 \log_7 4 = wrong answer! This flips the base and argument, giving you log⁡47 \log_4 7 instead. Always put the argument (4) in numerator and base (7) in denominator.

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 calculate log⁡74 \log_7 4 directly on my calculator?

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Most calculators only have common log (log₁₀) and natural log (ln) buttons. To find logs with other bases like 7, you must use the change of base formula.

Can I use log₁₀ instead of ln in the change of base formula?

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Absolutely! log⁡74=log⁡4log⁡7 \log_7 4 = \frac{\log 4}{\log 7} works just as well. Both natural logs and common logs give the same final answer.

How do I remember which number goes on top?

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Think of it as "what you want" over "what base you have". For log⁡74 \log_7 4 , you want 4, so it goes on top: ln⁡4ln⁡7 \frac{\ln 4}{\ln 7} .

What does log⁡74 \log_7 4 actually mean?

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It asks: "What power must I raise 7 to get 4?" Since 70.712≈4 7^{0.712} \approx 4 , the answer is approximately 0.712.

Why do some answer choices have different bases like log₅ or log₈?

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Those are incorrect applications of the change of base formula. Only ln⁡4ln⁡7 \frac{\ln 4}{\ln 7} correctly converts log⁡74 \log_7 4 using the standard logarithm bases.

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