Solve the Logarithmic Equation: (2log₃2 + log₃x)log₂3 - log₂x = 3x-7

Q: Why do I need to use the change of base formula here?

The equation has both $ \log_3 $ and $ \log_2 $ terms mixed together. You can't combine or simplify logarithms with different bases. The change of base formula lets you convert everything to one base so terms can cancel out.

Q: How do I remember the change of base formula?

Think of it as a fraction flip! $ \log_a b = \frac{1}{\log_b a} $. So $ \log_2 3 = \frac{1}{\log_3 2} $. The bases swap positions and become a reciprocal.

Q: Why do the logarithmic terms cancel out completely?

After converting to the same base, we get $ \frac{\log_3 x}{\log_3 2} - \frac{\log_3 x}{\log_3 2} = 0 $. These are identical terms with opposite signs, so they cancel perfectly, leaving just the constant terms.

Q: What if I chose base 2 instead of base 3?

You'd get the same answer! The key is being consistent. Converting everything to base 2 would also make the logarithmic terms cancel, leaving the same linear equation to solve.

Q: How can I verify my answer is correct?

Substitute $ x = 3 $ back into the original equation. Calculate each side separately: the left side simplifies to 2, and the right side $ 3(3) - 7 = 2 $. Both sides equal 2, confirming our solution!

Q: Are there always solutions to logarithmic equations?

No! Logarithmic equations can have no solution, one solution, or multiple solutions. Always check that your answer makes all logarithms defined (arguments must be positive) and satisfies the original equation.

Logarithmic Equations with Change of Base

$(2\log_32+\log_3x)\log_23-\log_2x=3x-7$

$x=\text{?}$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:13 Let's find the value of X.

00:18 To do this, we'll use the power log formula. First, move the coefficient into the log.

00:30 Now, let's apply this formula to our exercise.

00:50 Next, we'll use the addition formula. It gets the log of their product.

00:55 We'll apply this formula in the exercise.

01:16 Now, we'll use the multiplication formula to switch between numbers.

01:31 We'll apply this formula in the exercise.

01:46 Remember: The log of a number in its own base is always one.

01:51 We'll use this rule in our exercise.

02:06 Next, use the subtraction formula for the log of their quotient.

02:16 We'll apply this formula in the exercise.

02:31 Let's calculate the log and then substitute the values.

02:51 Isolate X and solve to find our solution.

03:01 And that's how we solve the problem!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

$(2\log_32+\log_3x)\log_23-\log_2x=3x-7$

$x=\text{?}$

Step-by-step solution

Let's solve the given equation step by step:

We start with:

$(2\log_3 2 + \log_3 x)\log_2 3 - \log_2 x = 3x - 7$

Firstly, use the change of base formula to convert $\log_2 3$ to base 3:

$\log_2 3 = \frac{\log_3 3}{\log_3 2} = \frac{1}{\log_3 2}$

Substitute this expression into the original equation:

$(2\log_3 2 + \log_3 x)\left(\frac{1}{\log_3 2}\right) - \log_2 x = 3x - 7$

Simplify the first term:

$\frac{2\log_3 2 + \log_3 x}{\log_3 2} = 2 + \frac{\log_3 x}{\log_3 2}$

Thus, the equation becomes:

$2 + \frac{\log_3 x}{\log_3 2} - \log_2 x = 3x - 7$

Convert $\log_2 x$ to base 3 using change of base:

$\log_2 x = \frac{\log_3 x}{\log_3 2}$

Substitute back into the equation:

$2 + \frac{\log_3 x}{\log_3 2} - \frac{\log_3 x}{\log_3 2} = 3x - 7$

The middle terms cancel out, simplifying to:

2 = 3x - 7

Solving for $x$ :

Add 7 to both sides:

$9 = 3x$

Divide by 3:

$x = 3$

Thus, the solution to the problem is $x = 3$ .

Final Answer

$3$

Key Points to Remember

Essential concepts to master this topic

Change of Base Formula: Convert all logarithms to same base
Technique: Use $\log_2 3 = \frac{1}{\log_3 2}$ to simplify expressions
Check: Substitute x = 3 into original equation: both sides equal 2 ✓

Common Mistakes

Avoid these frequent errors

Forgetting to apply change of base consistently
Don't convert only some logarithms while leaving others in different bases = mixed bases that can't be simplified! This creates impossible-to-solve expressions. Always convert all logarithms to the same base before attempting algebraic manipulation.

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 I need to use the change of base formula here?

The equation has both $\log_3$ and $\log_2$ terms mixed together. You can't combine or simplify logarithms with different bases. The change of base formula lets you convert everything to one base so terms can cancel out.

How do I remember the change of base formula?

Think of it as a fraction flip! $\log_a b = \frac{1}{\log_b a}$ . So $\log_2 3 = \frac{1}{\log_3 2}$ . The bases swap positions and become a reciprocal.

Why do the logarithmic terms cancel out completely?

After converting to the same base, we get $\frac{\log_3 x}{\log_3 2} - \frac{\log_3 x}{\log_3 2} = 0$ . These are identical terms with opposite signs, so they cancel perfectly, leaving just the constant terms.

What if I chose base 2 instead of base 3?

You'd get the same answer! The key is being consistent. Converting everything to base 2 would also make the logarithmic terms cancel, leaving the same linear equation to solve.

How can I verify my answer is correct?

Substitute $x = 3$ back into the original equation. Calculate each side separately: the left side simplifies to 2, and the right side $3(3) - 7 = 2$ . Both sides equal 2, confirming our solution!

Are there always solutions to logarithmic equations?

No! Logarithmic equations can have no solution, one solution, or multiple solutions. Always check that your answer makes all logarithms defined (arguments must be positive) and satisfies the original equation.

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Solve the Logarithmic Equation: (2log₃2 + log₃x)log₂3 - log₂x = 3x-7

Logarithmic Equations with Change of Base

❤️ 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 I need to use the change of base formula here?

How do I remember the change of base formula?

Why do the logarithmic terms cancel out completely?

What if I chose base 2 instead of base 3?

How can I verify my answer is correct?

Are there always solutions to logarithmic equations?

🌟 Unlock Your Math Potential

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