Math / Algebra

QuestionSolve for $x$ . $4^{x+3}=3^{2 x}$
Write the exact answer using either base-10 or base-e logarithms. $x=$ $\square$ $\square \log$ 口 ㅁIn $\square$ $\square$ $\square$ No solution

Studdy Solution

STEP 1

What is this asking? We've got an equation with the variable $x$ stuck in the exponents of both $4$ and $3$ , and we need to free it! Watch out! It's tempting to think $4$ and $3$ are related somehow, but they're not cooperating today.
We'll need logarithms to bring those exponents down.

STEP 2

1. Apply Logarithms
2. Expand and Rearrange
3. Isolate and Solve

STEP 3

Let's take the logarithm of both sides of our equation $4^{x+3} = 3^{2x}$ .
We can use any logarithm we want, but the common logarithm (base 10) is a good choice.
So, we have $\log(4^{x+3}) = \log(3^{2x})$ .
Why? Because if two things are equal, then their logarithms are equal too!

STEP 4

Now, remember the power rule of logarithms: $\log(a^b) = b \cdot \log(a)$ .
This rule is our key to getting $x$ out of those exponents!
Applying this rule to both sides, we get $(x+3) \cdot \log(4) = 2x \cdot \log(3)$ .

STEP 5

Let's distribute the logarithms on the left side: $x \cdot \log(4) + 3 \cdot \log(4) = 2x \cdot \log(3)$ .
We're getting closer to isolating $x$ !

STEP 6

Now, we want to group all the terms with $x$ on one side and all the terms without $x$ on the other.
Let's subtract $x \cdot \log(4)$ from both sides: $3 \cdot \log(4) = 2x \cdot \log(3) - x \cdot \log(4)$ .

STEP 7

We can factor out $x$ on the right side: $3 \cdot \log(4) = x \cdot (2 \cdot \log(3) - \log(4))$ .

STEP 8

Finally, let's divide both sides by $(2 \cdot \log(3) - \log(4))$ to isolate $x$ : $x = \frac{3 \cdot \log(4)}{2 \cdot \log(3) - \log(4)}$

STEP 9

Now, we just need to plug this into a calculator to get our final result: $x = \frac{3 \cdot \log(4)}{2 \cdot \log(3) - \log(4)} \approx \frac{3 \cdot 0.602}{2 \cdot 0.477 - 0.602} \approx \frac{1.806}{0.954 - 0.602} \approx \frac{1.806}{0.352} \approx 5.13$ So, $x \approx 5.13$ .

STEP 10

$x = \frac{3 \cdot \log(4)}{2 \cdot \log(3) - \log(4)}$

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QuestionSolve for $x$ . $4^{x+3}=3^{2 x}$
Write the exact answer using either base-10 or base-e logarithms. $x=$ $\square$ $\square \log$ 口 ㅁIn $\square$ $\square$ $\square$ No solution

Studdy Solution

STEP 1

What is this asking? We've got an equation with the variable $x$ stuck in the exponents of both $4$ and $3$ , and we need to free it! Watch out! It's tempting to think $4$ and $3$ are related somehow, but they're not cooperating today.
We'll need logarithms to bring those exponents down.

STEP 2

1. Apply Logarithms
2. Expand and Rearrange
3. Isolate and Solve

STEP 3

Let's take the logarithm of both sides of our equation $4^{x+3} = 3^{2x}$ .
We can use any logarithm we want, but the common logarithm (base 10) is a good choice.
So, we have $\log(4^{x+3}) = \log(3^{2x})$ .
Why? Because if two things are equal, then their logarithms are equal too!

STEP 4

Now, remember the power rule of logarithms: $\log(a^b) = b \cdot \log(a)$ .
This rule is our key to getting $x$ out of those exponents!
Applying this rule to both sides, we get $(x+3) \cdot \log(4) = 2x \cdot \log(3)$ .

STEP 5

Let's distribute the logarithms on the left side: $x \cdot \log(4) + 3 \cdot \log(4) = 2x \cdot \log(3)$ .
We're getting closer to isolating $x$ !

STEP 6

Now, we want to group all the terms with $x$ on one side and all the terms without $x$ on the other.
Let's subtract $x \cdot \log(4)$ from both sides: $3 \cdot \log(4) = 2x \cdot \log(3) - x \cdot \log(4)$ .

STEP 7

We can factor out $x$ on the right side: $3 \cdot \log(4) = x \cdot (2 \cdot \log(3) - \log(4))$ .

STEP 8

Finally, let's divide both sides by $(2 \cdot \log(3) - \log(4))$ to isolate $x$ : $x = \frac{3 \cdot \log(4)}{2 \cdot \log(3) - \log(4)}$

STEP 9

Now, we just need to plug this into a calculator to get our final result: $x = \frac{3 \cdot \log(4)}{2 \cdot \log(3) - \log(4)} \approx \frac{3 \cdot 0.602}{2 \cdot 0.477 - 0.602} \approx \frac{1.806}{0.954 - 0.602} \approx \frac{1.806}{0.352} \approx 5.13$ So, $x \approx 5.13$ .

STEP 10

$x = \frac{3 \cdot \log(4)}{2 \cdot \log(3) - \log(4)}$

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QuestionSolve for xxx. 4x+3=32x4^{x+3}=3^{2 x}4x+3=32xWrite the exact answer using either base-10 or base-e logarithms. x=x=x= □\square□ □log⁡\square \log□log 口 ㅁIn □\square□ □\square□ □\square□ No solution

Studdy Solution

STEP 1

What is this asking? We've got an equation with the variable xxx stuck in the exponents of both 444 and 333, and we need to free it! Watch out! It's tempting to think 444 and 333 are related somehow, but they're not cooperating today.We'll need logarithms to bring those exponents down.

STEP 2

1. Apply Logarithms2. Expand and Rearrange3. Isolate and Solve

STEP 3

STEP 4

STEP 5

Let's **distribute** the logarithms on the left side: x⋅log⁡(4)+3⋅log⁡(4)=2x⋅log⁡(3)x \cdot \log(4) + 3 \cdot \log(4) = 2x \cdot \log(3)x⋅log(4)+3⋅log(4)=2x⋅log(3).We're getting closer to isolating xxx!

STEP 6

STEP 7

We can **factor out** xxx on the right side: 3⋅log⁡(4)=x⋅(2⋅log⁡(3)−log⁡(4))3 \cdot \log(4) = x \cdot (2 \cdot \log(3) - \log(4))3⋅log(4)=x⋅(2⋅log(3)−log(4)).

STEP 8

Finally, let's **divide both sides** by (2⋅log⁡(3)−log⁡(4))(2 \cdot \log(3) - \log(4))(2⋅log(3)−log(4)) to isolate xxx: x=3⋅log⁡(4)2⋅log⁡(3)−log⁡(4)x = \frac{3 \cdot \log(4)}{2 \cdot \log(3) - \log(4)}x=2⋅log(3)−log(4)3⋅log(4)​

STEP 9

STEP 10

x=3⋅log⁡(4)2⋅log⁡(3)−log⁡(4)x = \frac{3 \cdot \log(4)}{2 \cdot \log(3) - \log(4)}x=2⋅log(3)−log(4)3⋅log(4)​

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Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

QuestionSolve for $x$ . $4^{x+3}=3^{2 x}$
Write the exact answer using either base-10 or base-e logarithms. $x=$ $\square$ $\square \log$ 口 ㅁIn $\square$ $\square$ $\square$ No solution

What is this asking? We've got an equation with the variable $x$ stuck in the exponents of both $4$ and $3$ , and we need to free it! Watch out! It's tempting to think $4$ and $3$ are related somehow, but they're not cooperating today.
We'll need logarithms to bring those exponents down.

1. Apply Logarithms
2. Expand and Rearrange
3. Isolate and Solve

Let's distribute the logarithms on the left side: $x \cdot \log(4) + 3 \cdot \log(4) = 2x \cdot \log(3)$ .
We're getting closer to isolating $x$ !

We can factor out $x$ on the right side: $3 \cdot \log(4) = x \cdot (2 \cdot \log(3) - \log(4))$ .

Finally, let's divide both sides by $(2 \cdot \log(3) - \log(4))$ to isolate $x$ : $x = \frac{3 \cdot \log(4)}{2 \cdot \log(3) - \log(4)}$

$x = \frac{3 \cdot \log(4)}{2 \cdot \log(3) - \log(4)}$