Solve the Logarithmic Equation: log₇(x⁴) - log₇(2x²) = 3

Q: Why can I use the quotient rule here?

The quotient rule applies because we have subtraction of logarithms with the same base: $ \log_7 x^4 - \log_7 2x^2 $. This becomes $ \log_7 \frac{x^4}{2x^2} $.

Q: How do I simplify the fraction inside the logarithm?

Simplify $ \frac{x^4}{2x^2} $ by canceling common factors: $ \frac{x^4}{2x^2} = \frac{x^2}{2} $. Remember that $ x^4 ÷ x^2 = x^2 $!

Q: What does it mean when log₇(something) = 3?

This means 7 raised to the power of 3 equals that something. So $ \log_7 \frac{x^2}{2} = 3 $ becomes $ 7^3 = \frac{x^2}{2} $, which is $ 343 = \frac{x^2}{2} $.

Q: Why do I get two answers from one equation?

Because we end up with $ x^2 = 686 $, and every positive number has two square roots: one positive and one negative. Both $ 7\sqrt{14} $ and $ -7\sqrt{14} $ work!

Q: How do I check if negative solutions work in logarithms?

Even though $ x = -7\sqrt{14} $ is negative, when we substitute it back, we get $ x^4 = (-7\sqrt{14})^4 $ and $ x^2 = (-7\sqrt{14})^2 $, which are both positive, so the logarithms are defined!

Logarithmic Properties with Exponential Solutions

$\log_7x^4-\log_72x^2=3$

?=x

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Solve

00:05 We'll use the logarithm subtraction formula, we'll get the logarithm of their quotient

00:12 We'll use this formula in our exercise

00:26 Let's simplify what we can

00:37 Solve according to the logarithm definition

00:45 Isolate X

01:11 When extracting a root there are always 2 solutions, positive and negative

01:16 These are the possible solutions

01:29 Let's check the domain

01:34 According to the domain we'll find the solution

01:41 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

$\log_7x^4-\log_72x^2=3$

?=x

Step-by-step solution

$\log_ax-\log_ay=\log_a\frac{x}{y}$

$\log_7x^4-\log_72x^2=$

$\log_7\frac{x^4}{2x^2}=3$

$7^3=\frac{x^2}{2}$

We multiply by: $2$

$2\cdot7^3=x^2$

Extract the root

$x=\sqrt{680}=7\sqrt{14}$

$x=-\sqrt{680}=-7\sqrt{14}$

Final Answer

$-7\sqrt{14\text{ }}\text{ , }7\sqrt{14}$

Key Points to Remember

Essential concepts to master this topic

Property: Use quotient rule: $\log_a x - \log_a y = \log_a \frac{x}{y}$
Technique: Simplify $\frac{x^4}{2x^2} = \frac{x^2}{2}$ before solving exponentially
Check: Substitute both solutions back: $\log_7(±7\sqrt{14})^4 - \log_7(2(±7\sqrt{14})^2) = 3$ ✓

Common Mistakes

Avoid these frequent errors

Forgetting the negative square root solution
Don't solve $x^2 = 686$ and only take $x = 7\sqrt{14}$ = missing half the solutions! When you square root both sides, you get both positive AND negative values. Always include both $x = ±7\sqrt{14}$ as your final answer.

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 I use the quotient rule here?

The quotient rule applies because we have subtraction of logarithms with the same base: $\log_7 x^4 - \log_7 2x^2$ . This becomes $\log_7 \frac{x^4}{2x^2}$ .

How do I simplify the fraction inside the logarithm?

Simplify $\frac{x^4}{2x^2}$ by canceling common factors: $\frac{x^4}{2x^2} = \frac{x^2}{2}$ . Remember that $x^4 ÷ x^2 = x^2$ !

What does it mean when log₇(something) = 3?

This means 7 raised to the power of 3 equals that something. So $\log_7 \frac{x^2}{2} = 3$ becomes $7^3 = \frac{x^2}{2}$ , which is $343 = \frac{x^2}{2}$ .

Why do I get two answers from one equation?

Because we end up with $x^2 = 686$ , and every positive number has two square roots: one positive and one negative. Both $7\sqrt{14}$ and $-7\sqrt{14}$ work!

How do I check if negative solutions work in logarithms?

Even though $x = -7\sqrt{14}$ is negative, when we substitute it back, we get $x^4 = (-7\sqrt{14})^4$ and $x^2 = (-7\sqrt{14})^2$ , which are both positive, so the logarithms are defined!

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Solve the Logarithmic Equation: log₇(x⁴) - log₇(2x²) = 3

Logarithmic Properties with Exponential Solutions

❤️ 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 I use the quotient rule here?

How do I simplify the fraction inside the logarithm?

What does it mean when log₇(something) = 3?

Why do I get two answers from one equation?

How do I check if negative solutions work in logarithms?

🌟 Unlock Your Math Potential

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