Solve 2log₃8: Step-by-Step Logarithm Equation Solution

Q: Why does the coefficient 2 become an exponent?

This comes from the power property of logarithms: $ a \cdot \log_b c = \log_b(c^a) $. The coefficient always moves inside as an exponent on the argument, not the base!

Q: What if I have a fraction as the coefficient?

Same rule applies! For example, $ \frac{1}{2}\log_3 8 = \log_3(8^{1/2}) = \log_3(\sqrt{8}) $. The fraction becomes the exponent inside.

Q: Can I work backwards from the answer choices?

Yes! Since $ \log_3 64 $ means "3 to what power equals 64?", you can verify this equals $ 2\log_3 8 $ using the power property.

Q: How do I remember which direction the power property goes?

Think: "Coefficient goes inside as an exponent." The number outside the log always moves to become an exponent on the argument inside the parentheses.

Q: What if the base changes in the answer choices?

Be careful! The base never changes when using the power property. If you see $ \log_6 8 $ as an option, it's likely a trap answer.

Logarithm Properties with Power Rules

$2\log_38=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Solve

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

00:11 We'll use this formula in our exercise

00:21 Let's calculate the power

00:25 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

$2\log_38=$

Step-by-step solution

To solve this problem, let's simplify $2\log_3 8$ using logarithm rules.

Step 1: Recognize the expression form
The expression is of the form $a \cdot \log_b c$ , where $a = 2$ , $b = 3$ , and $c = 8$ .
Step 2: Apply the power property
According to the power property of logarithms, $2 \cdot \log_3 8$ can be simplified to $\log_3 (8^2)$ .
Perform the calculation
Calculate $8^2$ , which is $64$ .
Step 3: Simplify further
Therefore, we have $\log_3 64$ .

This is a straightforward application of the power property of logarithms. By applying this property correctly, we've simplified the original expression correctly.

Therefore, the simplified form of $2\log_3 8$ is $\log_3 64$ .

Final Answer

$\log_364$

Key Points to Remember

Essential concepts to master this topic

Power Property: Coefficient becomes exponent inside logarithm argument
Technique: Transform $2\log_3 8$ to $\log_3(8^2)$ then calculate $8^2 = 64$
Check: Verify $2\log_3 8 = \log_3 64$ using calculator approximations ✓

Common Mistakes

Avoid these frequent errors

Moving coefficient to the base instead of argument
Don't change $2\log_3 8$ to $\log_6 8$ = wrong base! The coefficient multiplies the entire logarithm, not the base. Always move the coefficient as an exponent to the argument inside the log.

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 coefficient 2 become an exponent?

This comes from the power property of logarithms: $a \cdot \log_b c = \log_b(c^a)$ . The coefficient always moves inside as an exponent on the argument, not the base!

What if I have a fraction as the coefficient?

Same rule applies! For example, $\frac{1}{2}\log_3 8 = \log_3(8^{1/2}) = \log_3(\sqrt{8})$ . The fraction becomes the exponent inside.

Can I work backwards from the answer choices?

Yes! Since $\log_3 64$ means "3 to what power equals 64?", you can verify this equals $2\log_3 8$ using the power property.

How do I remember which direction the power property goes?

Think: "Coefficient goes inside as an exponent." The number outside the log always moves to become an exponent on the argument inside the parentheses.

What if the base changes in the answer choices?

Be careful! The base never changes when using the power property. If you see $\log_6 8$ as an option, it's likely a trap answer.

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Solve 2log₃8: Step-by-Step Logarithm Equation Solution

Logarithm Properties with Power Rules

❤️ 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 coefficient 2 become an exponent?

What if I have a fraction as the coefficient?

Can I work backwards from the answer choices?

How do I remember which direction the power property goes?

What if the base changes in the answer choices?

🌟 Unlock Your Math Potential

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