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

Q: Why can't x ln(7) be written as ln(7x)?

Because ln(7x) means the natural log of 7 times x, while x ln(7) means x times the natural log of 7. These are completely different expressions! The power rule transforms x ln(7) into ln(7^x).

Q: How do I remember the power rule for logarithms?

Think of it as moving the coefficient: when you have b ln(a), the coefficient b becomes the exponent, giving you ln(a^b). The coefficient 'jumps up' to become the power!

Q: What's the difference between ln(7^x) and 7 ln(x)?

ln(7^x) is the natural log of 7 raised to the x power, while 7 ln(x) is 7 times the natural log of x. Only ln(7^x) equals x ln(7) by the power rule.

Q: Can I use this rule with any logarithm base?

Yes! The power rule works for all logarithm bases. Whether it's natural log (ln), common log (log), or any other base, the rule b log(a) = log(a^b) always applies.

Q: How can I check if I applied the power rule correctly?

Start with your answer and expand it backIf ln(7^x) is correct, it should expand to x ln(7)Use the power rule in reverse to verify: ln(7^x) = x ln(7) ✓

Logarithm Properties with Power Rule

$x\ln7=$

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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 to convert from ln to log

00:09 We will use this formula in our exercise

00:19 We will use the formula for the log of a power

00:29 We will use this formula in our exercise

00:40 We will convert back to ln again, using the formula

00:44 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

$x\ln7=$

Step-by-step solution

To solve this problem, we'll follow the steps outlined:

Step 1: Recognize that the expression $x \ln 7$ can be thought of in terms of the power property of logarithms, which helps reframe it into a single logarithm.
Step 2: Apply the formula $\ln(a^b) = b \ln a$ . This tells us that if we have something of the form $b \ln a$ , we can express it as $\ln(a^b)$ .
Step 3: Utilize the known expression and rule by substituting $a = 7$ and $b = x$ . Thus, $x \ln 7$ becomes $\ln(7^x)$ .

Therefore, the rewritten expression for $x \ln 7$ using logarithm rules is $\ln 7^x$ .

This matches choice 4 from the provided options.

Final Answer

$\ln7^x$

Key Points to Remember

Essential concepts to master this topic

Power Rule: The expression b ln(a) equals ln(a^b)
Technique: Transform x ln(7) by applying ln(7^x) using power property
Check: Verify ln(7^x) expands back to x ln(7) using logarithm rules ✓

Common Mistakes

Avoid these frequent errors

Confusing coefficient placement in logarithmic expressions
Don't write x ln(7) as ln(7x) or 7 ln(x) = wrong positioning! This changes the mathematical meaning completely and gives incorrect results. Always apply the power rule: coefficient times logarithm becomes logarithm of base raised to that power.

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 x ln(7) be written as ln(7x)?

Because ln(7x) means the natural log of 7 times x, while x ln(7) means x times the natural log of 7. These are completely different expressions! The power rule transforms x ln(7) into ln(7^x).

How do I remember the power rule for logarithms?

Think of it as moving the coefficient: when you have b ln(a), the coefficient b becomes the exponent, giving you ln(a^b). The coefficient 'jumps up' to become the power!

What's the difference between ln(7^x) and 7 ln(x)?

ln(7^x) is the natural log of 7 raised to the x power, while 7 ln(x) is 7 times the natural log of x. Only ln(7^x) equals x ln(7) by the power rule.

Can I use this rule with any logarithm base?

Yes! The power rule works for all logarithm bases. Whether it's natural log (ln), common log (log), or any other base, the rule b log(a) = log(a^b) always applies.

How can I check if I applied the power rule correctly?

Start with your answer and expand it back
If ln(7^x) is correct, it should expand to x ln(7)
Use the power rule in reverse to verify: ln(7^x) = x ln(7) ✓

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

Logarithm Properties with Power 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 x ln(7) be written as ln(7x)?

How do I remember the power rule for logarithms?

What's the difference between ln(7^x) and 7 ln(x)?

Can I use this rule with any logarithm base?

How can I check if I applied the power rule correctly?

🌟 Unlock Your Math Potential

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