Solve the Logarithmic Equation: 3log₇6 Step by Step

Q: Why does the 3 become an exponent instead of multiplying the base?

The power property states that $ a\log_b c = \log_b(c^a) $. The coefficient always becomes an exponent of the argument (the number being logged), never the base!

Q: What's the difference between $ 3\log_76 $ and $ \log_3 6 $ ?

These are completely different! $ 3\log_76 $ means "3 times log base 7 of 6" while $ \log_3 6 $ means "log base 3 of 6". The position of the number matters!

Q: Do I always have to calculate the exponent like $ 6^3 = 216 $ ?

Not always! Sometimes leaving it as $ \log_7(6^3) $ is the final answer. But when answer choices show specific numbers like 216, you should calculate the exponent to match the format.

Q: Can I use this property backwards?

Yes! You can convert $ \log_7216 $ back to $ 3\log_76 $ if you recognize that $ 216 = 6^3 $. This is useful for simplifying complex logarithmic expressions.

Q: What if the coefficient is a fraction like $ \frac{1}{2}\log_76 $ ?

The same rule applies! $ \frac{1}{2}\log_76 = \log_7(6^{1/2}) = \log_7\sqrt{6} $. Fractional exponents create roots inside the logarithm.

Logarithm Properties with Power Coefficients

$3\log_76=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Solve

00:03 We will use the formula for the logarithm of a power

00:16 We will use this formula in our exercise

00:23 Let's calculate the power

00:26 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

$3\log_76=$

Step-by-step solution

To simplify the expression $3\log_76$ , we apply the power property of logarithms, which states:

$a\log_b c = \log_b(c^a)$

Step 1: Identify the given expression: $3\log_76$ .

Step 2: Apply the power property of logarithms:

$3\log_76 = \log_7(6^3)$

Step 3: Calculate $6^3$ :

$6^3 = 6 \times 6 \times 6 = 36 \times 6 = 216$

Step 4: Substitute back into the logarithmic expression:

$\log_7(6^3) = \log_7216$

Therefore, the simplified expression is $\log_7216$ .

Comparing with the answer choices, the correct choice is:

$\log_7216$

Final Answer

$\log_7216$

Key Points to Remember

Essential concepts to master this topic

Power Rule: Coefficient becomes exponent inside the logarithm
Technique: Convert $3\log_76$ to $\log_7(6^3)$
Check: Verify $6^3 = 216$ gives $\log_7216$ ✓

Common Mistakes

Avoid these frequent errors

Adding coefficient to base or argument
Don't change $3\log_76$ to $\log_{10}6$ or $\log_79$ = wrong property applied! The coefficient multiplies the entire logarithm, not individual parts. Always use the power property: coefficient becomes an exponent of the argument.

Practice Quiz

Test your knowledge with interactive questions

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

FAQ

Everything you need to know about this question

Why does the 3 become an exponent instead of multiplying the base?

The power property states that $a\log_b c = \log_b(c^a)$ . The coefficient always becomes an exponent of the argument (the number being logged), never the base!

What's the difference between $3\log_76$ and $\log_3 6$ ?

These are completely different! $3\log_76$ means "3 times log base 7 of 6" while $\log_3 6$ means "log base 3 of 6". The position of the number matters!

Do I always have to calculate the exponent like $6^3 = 216$ ?

Not always! Sometimes leaving it as $\log_7(6^3)$ is the final answer. But when answer choices show specific numbers like 216, you should calculate the exponent to match the format.

Can I use this property backwards?

Yes! You can convert $\log_7216$ back to $3\log_76$ if you recognize that $216 = 6^3$ . This is useful for simplifying complex logarithmic expressions.

What if the coefficient is a fraction like $\frac{1}{2}\log_76$ ?

The same rule applies! $\frac{1}{2}\log_76 = \log_7(6^{1/2}) = \log_7\sqrt{6}$ . Fractional exponents create roots inside the logarithm.

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Solve the Logarithmic Equation: 3log₇6 Step by Step

Logarithm Properties with Power Coefficients

❤️ 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 does the 3 become an exponent instead of multiplying the base?

What's the difference between $3\log_76$ and $\log_3 6$ ?

Do I always have to calculate the exponent like $6^3 = 216$ ?

Can I use this property backwards?

What if the coefficient is a fraction like $\frac{1}{2}\log_76$ ?

🌟 Unlock Your Math Potential

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Solve the Logarithmic Equation: 3log₇6 Step by Step

Logarithm Properties with Power Coefficients

❤️ 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 does the 3 become an exponent instead of multiplying the base?

What's the difference between 3log⁡76 3\log_76 3log7​6 and log⁡36 \log_3 6 log3​6?

Do I always have to calculate the exponent like 63=216 6^3 = 216 63=216?

Can I use this property backwards?

What if the coefficient is a fraction like 12log⁡76 \frac{1}{2}\log_76 21​log7​6?

🌟 Unlock Your Math Potential

More Questions

Rules of Logarithms

Solve for X: Finding the Value in x ln(7) Equation

Solve the Logarithmic Fraction: log₈5 ÷ log₈9

Solve Complex Logarithm Equation: log₂ₐ(e⁷) with Natural Logarithms

Solve the Complex Logarithmic Equation: log₅x + log₅(x+2) + log₂5 - log₂2.5 = log₃7 × log₇9

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Solve log₄(9x) + log₄(x+4) - log₄(3) = ln(2e) + ln(1/2e): Finding X

Solve the Logarithmic Equation: ln x + ln(x+1) - ln2 = 3

What's the difference between $3\log_76$ and $\log_3 6$ ?

Do I always have to calculate the exponent like $6^3 = 216$ ?

What if the coefficient is a fraction like $\frac{1}{2}\log_76$ ?