Math / Algebra

QuestionSolve the exporential equation. Expresi irrational solutions as decimals correct to the vearest thovianath. $5^{x-2}=3^{2 x}$
Select the correct choice A. The solution set is $\}$ B. empty set

Studdy Solution

STEP 1

What is this asking? We need to find the value of $x$ that makes $5^{x-2}$ equal to $3^{2x}$ , and if the answer isn't a nice whole number, we'll round it to the nearest thousandth. Watch out! Exponent rules are key here, so let's make sure we don't mix them up!
Also, remember to round carefully at the end.

STEP 2

1. Apply Natural Logarithm
2. Expand Using Logarithm Rules
3. Isolate x
4. Calculate and Round

STEP 3

We start with our equation: $5^{x-2} = 3^{2x}$ Now, to bring those exponents down, we'll take the natural logarithm (ln) of both sides.
Remember, what we do to one side, we must do to the other! $\ln(5^{x-2}) = \ln(3^{2x})$ Why is this a good idea?
Because it lets us use our awesome logarithm power rule in the next step!

STEP 4

Remember that handy logarithm power rule?
It says $\ln(a^b) = b \cdot \ln(a)$ .
Let's use it! $(x-2) \cdot \ln(5) = 2x \cdot \ln(3)$ Now, those exponents aren't stuck up there anymore, making things much easier to manage.

STEP 5

Let's distribute $\ln(5)$ on the left side: $x \cdot \ln(5) - 2 \cdot \ln(5) = 2x \cdot \ln(3)$ This helps us separate the terms with $x$ from the constant terms.

STEP 6

We want to get all the terms with $x$ on one side and everything else on the other.
Let's subtract $x \cdot \ln(5)$ from both sides: $-2 \cdot \ln(5) = 2x \cdot \ln(3) - x \cdot \ln(5)$ This groups our $x$ terms together, making it easier to solve for $x$ .

STEP 7

Now, we can factor out $x$ from the right side: $-2 \cdot \ln(5) = x \cdot (2 \cdot \ln(3) - \ln(5))$ This isolates $x$ , which is exactly what we want!

STEP 8

Finally, we can divide both sides by $(2 \cdot \ln(3) - \ln(5))$ to solve for $x$ : $x = \frac{-2 \cdot \ln(5)}{2 \cdot \ln(3) - \ln(5)}$ We've got $x$ all by itself now!

STEP 9

Using a calculator, we find: $\ln(5) \approx 1.6094$ $\ln(3) \approx 1.0986$ So, $x \approx \frac{-2 \cdot 1.6094}{2 \cdot 1.0986 - 1.6094} \approx \frac{-3.2188}{2.1972 - 1.6094} \approx \frac{-3.2188}{0.5878} \approx -5.4781$

STEP 10

Rounding to the nearest thousandth gives us: $x \approx -5.478$

STEP 11

The solution set is $\{-5.478\}$ .
So the correct choice is A.

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QuestionSolve the exporential equation. Expresi irrational solutions as decimals correct to the vearest thovianath. $5^{x-2}=3^{2 x}$
Select the correct choice A. The solution set is $\}$ B. empty set

Studdy Solution

STEP 1

STEP 2

1. Apply Natural Logarithm
2. Expand Using Logarithm Rules
3. Isolate x
4. Calculate and Round

STEP 3

STEP 4

Remember that handy logarithm power rule?
It says $\ln(a^b) = b \cdot \ln(a)$ .
Let's use it! $(x-2) \cdot \ln(5) = 2x \cdot \ln(3)$ Now, those exponents aren't stuck up there anymore, making things much easier to manage.

STEP 5

Let's distribute $\ln(5)$ on the left side: $x \cdot \ln(5) - 2 \cdot \ln(5) = 2x \cdot \ln(3)$ This helps us separate the terms with $x$ from the constant terms.

STEP 6

We want to get all the terms with $x$ on one side and everything else on the other.
Let's subtract $x \cdot \ln(5)$ from both sides: $-2 \cdot \ln(5) = 2x \cdot \ln(3) - x \cdot \ln(5)$ This groups our $x$ terms together, making it easier to solve for $x$ .

STEP 7

Now, we can factor out $x$ from the right side: $-2 \cdot \ln(5) = x \cdot (2 \cdot \ln(3) - \ln(5))$ This isolates $x$ , which is exactly what we want!

STEP 8

Finally, we can divide both sides by $(2 \cdot \ln(3) - \ln(5))$ to solve for $x$ : $x = \frac{-2 \cdot \ln(5)}{2 \cdot \ln(3) - \ln(5)}$ We've got $x$ all by itself now!

STEP 9

Using a calculator, we find: $\ln(5) \approx 1.6094$ $\ln(3) \approx 1.0986$ So, $x \approx \frac{-2 \cdot 1.6094}{2 \cdot 1.0986 - 1.6094} \approx \frac{-3.2188}{2.1972 - 1.6094} \approx \frac{-3.2188}{0.5878} \approx -5.4781$

STEP 10

Rounding to the nearest thousandth gives us: $x \approx -5.478$

STEP 11

The solution set is $\{-5.478\}$ .
So the correct choice is A.

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QuestionSolve the exporential equation. Expresi irrational solutions as decimals correct to the vearest thovianath. 5x−2=32x5^{x-2}=3^{2 x}5x−2=32xSelect the correct choice A. The solution set is }\}} B. empty set

Studdy Solution

STEP 1

STEP 2

1. Apply Natural Logarithm2. Expand Using Logarithm Rules3. Isolate *x*4. Calculate and Round

STEP 3

STEP 4

STEP 5

Let's distribute ln⁡(5)\ln(5)ln(5) on the left side: x⋅ln⁡(5)−2⋅ln⁡(5)=2x⋅ln⁡(3)x \cdot \ln(5) - 2 \cdot \ln(5) = 2x \cdot \ln(3)x⋅ln(5)−2⋅ln(5)=2x⋅ln(3) This helps us separate the terms with xxx from the constant terms.

STEP 6

STEP 7

Now, we can factor out xxx from the right side: −2⋅ln⁡(5)=x⋅(2⋅ln⁡(3)−ln⁡(5))-2 \cdot \ln(5) = x \cdot (2 \cdot \ln(3) - \ln(5))−2⋅ln(5)=x⋅(2⋅ln(3)−ln(5)) This isolates xxx, which is exactly what we want!

STEP 8

Finally, we can divide both sides by (2⋅ln⁡(3)−ln⁡(5))(2 \cdot \ln(3) - \ln(5))(2⋅ln(3)−ln(5)) to solve for xxx: x=−2⋅ln⁡(5)2⋅ln⁡(3)−ln⁡(5)x = \frac{-2 \cdot \ln(5)}{2 \cdot \ln(3) - \ln(5)}x=2⋅ln(3)−ln(5)−2⋅ln(5)​ We've got xxx all by itself now!

STEP 9

STEP 10

Rounding to the nearest thousandth gives us: x≈−5.478x \approx -5.478x≈−5.478

STEP 11

The solution set is {−5.478}\{-5.478\}{−5.478}.So the correct choice is A.

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QuestionSolve the exporential equation. Expresi irrational solutions as decimals correct to the vearest thovianath. $5^{x-2}=3^{2 x}$
Select the correct choice A. The solution set is $\}$ B. empty set

1. Apply Natural Logarithm
2. Expand Using Logarithm Rules
3. Isolate x
4. Calculate and Round

Let's distribute $\ln(5)$ on the left side: $x \cdot \ln(5) - 2 \cdot \ln(5) = 2x \cdot \ln(3)$ This helps us separate the terms with $x$ from the constant terms.

Now, we can factor out $x$ from the right side: $-2 \cdot \ln(5) = x \cdot (2 \cdot \ln(3) - \ln(5))$ This isolates $x$ , which is exactly what we want!

Finally, we can divide both sides by $(2 \cdot \ln(3) - \ln(5))$ to solve for $x$ : $x = \frac{-2 \cdot \ln(5)}{2 \cdot \ln(3) - \ln(5)}$ We've got $x$ all by itself now!

Rounding to the nearest thousandth gives us: $x \approx -5.478$

The solution set is $\{-5.478\}$ .
So the correct choice is A.