Solve the Complex Logarithmic Equation: (log_x4 + log_x30.25)/(x*log_x11) + x = 3

Q: Why do we multiply 4 × 30.25 instead of solving each logarithm separately?

The logarithm product rule says $ \log_x a + \log_x b = \log_x(ab) $. This simplifies our equation from two separate logs to just one: $ \log_x 121 $, making it much easier to solve!

Q: How do I know that 121 = 11²?

Calculate step by step: $ 4 \times 30.25 = 121 $. Then recognize that $ 11 \times 11 = 121 $, so $ 121 = 11^2 $. This connection helps us see that $ \log_x 121 = 2\log_x 11 $!

Q: How do I verify that x = 2 is correct?

Substitute back: $ \frac{\log_2 4 + \log_2 30.25}{2 \cdot \log_2 11} + 2 $. Since $ \log_2 4 = 2 $ and the entire fraction simplifies to 1, we get 1 + 2 = 3 ✓

Q: Can I use the change of base formula to solve this?

Yes! You can convert to common logarithms: $ \log_x a = \frac{\ln a}{\ln x} $. However, recognizing patterns like $ 121 = 11^2 $ and testing simple values like x = 2 is often faster and more intuitive.

Logarithmic Equations with Base Transformations

$\frac{\log_x4+\log_x30.25}{x\log_x11}+x=3$

$x=\text{?}$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find X

00:04 Find the domain

00:17 Use the formula for logical addition, we'll get the log of their product

00:22 Use the power rule for logarithms, move X into the log

00:32 Calculate the product

00:40 Use the formula for logarithmic division

00:44 Get the log of numerator with base denominator

00:59 Solve the log and substitute in the equation

01:14 Group terms and isolate X to find the solution

01:29 Factor the trinomial to find possible solutions

01:34 Check the domain to find the solution

01:37 And this is the solution to the problem

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

$\frac{\log_x4+\log_x30.25}{x\log_x11}+x=3$

$x=\text{?}$

Step-by-step solution

To solve this problem, we'll follow these steps:

Step 1: Simplify the logarithmic expression $\log_x4 + \log_x30.25$ .
Step 2: Use the change of base formula for logarithms.
Step 3: Substitute and solve for $x$ .

Now, let's work through each step:
Step 1: Simplify the logarithmic expression by using the property $\log_x4 + \log_x30.25 = \log_x(4 \times 30.25)$ .
Step 2: Calculate $4 \times 30.25 = 121$ , then express as $\log_x121$ .

Step 3: The equation becomes $\frac{\log_x121}{x\log_x11} + x = 3$ . We know $\log_x121 = 2$ when $x = 11$ , thus evaluate the expression with possible values.

Consider a simpler value for $x$ , like 2. calc $\log_2 4 = 2$ and $\log_2 121$ . Using the logarithmic laws further simplifies if appropriate, achieving solution $x = 2$ .

Therefore, the solution to the problem is $x = 2$ .

Final Answer

$2$

Key Points to Remember

Essential concepts to master this topic

Product Rule: Combine logarithms using $\log_x a + \log_x b = \log_x(ab)$
Technique: Simplify $4 \times 30.25 = 121 = 11^2$ for easier calculation
Check: Substitute x = 2 back: $\frac{\log_2 121}{2\log_2 11} + 2 = 3$ ✓

Common Mistakes

Avoid these frequent errors

Forgetting to combine logarithms before solving
Don't solve $\log_x 4$ and $\log_x 30.25$ separately = wrong intermediate values! This makes the equation much harder and leads to calculation errors. Always use the product rule to combine: $\log_x 4 + \log_x 30.25 = \log_x(4 \times 30.25) = \log_x 121$ first.

Practice Quiz

Test your knowledge with interactive questions

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

FAQ

Everything you need to know about this question

Why do we multiply 4 × 30.25 instead of solving each logarithm separately?

The logarithm product rule says $\log_x a + \log_x b = \log_x(ab)$ . This simplifies our equation from two separate logs to just one: $\log_x 121$ , making it much easier to solve!

How do I know that 121 = 11²?

Calculate step by step: $4 \times 30.25 = 121$ . Then recognize that $11 \times 11 = 121$ , so $121 = 11^2$ . This connection helps us see that $\log_x 121 = 2\log_x 11$ !

What if x = 11 doesn't work in the original equation?

That's exactly right! When we substitute x = 11, we get division by zero in the denominator. This tells us x = 11 is not a valid solution, so we must try other values like x = 2.

How do I verify that x = 2 is correct?

Substitute back: $\frac{\log_2 4 + \log_2 30.25}{2 \cdot \log_2 11} + 2$ . Since $\log_2 4 = 2$ and the entire fraction simplifies to 1, we get 1 + 2 = 3 ✓

Can I use the change of base formula to solve this?

Yes! You can convert to common logarithms: $\log_x a = \frac{\ln a}{\ln x}$ . However, recognizing patterns like $121 = 11^2$ and testing simple values like x = 2 is often faster and more intuitive.

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Solve the Complex Logarithmic Equation: (log_x4 + log_x30.25)/(x*log_x11) + x = 3

Logarithmic Equations with Base Transformations

❤️ 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 do we multiply 4 × 30.25 instead of solving each logarithm separately?

How do I know that 121 = 11²?

What if x = 11 doesn't work in the original equation?

How do I verify that x = 2 is correct?

Can I use the change of base formula to solve this?

🌟 Unlock Your Math Potential

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