Math / Algebra

QuestionSolve the logarithmic equation. $2 \log _{8} x=5.8$
Select the correct choice and fill in any answer box in your choice below. A. The solution is $x \approx$ $\square$ . (Type an integer or decimal rounded to three decimal places as needed. Use a comma to separate answers as needed.) B. There is no solution.

Studdy Solution

STEP 1

What is this asking? We need to find the value of $x$ that satisfies the logarithmic equation $2 \log _{8} x = 5.8$ . Watch out! Remember the properties of logarithms, especially how to change the base and convert between logarithmic and exponential forms!

STEP 2

1. Isolate the logarithm
2. Convert to exponential form
3. Calculate the result

STEP 3

We want to get $\log_8 x$ by itself.
To do that, we divide both sides of the equation $2 \log _{8} x = 5.8$ by $2$ .
This is because we're undoing the multiplication by $2$ .
Think of it like this: if $2$ times something is $5.8$ , then that something must be $5.8$ divided by $2$ .
This gives us: $\frac{2 \log _{8} x}{2} = \frac{5.8}{2}$ $\log _{8} x = 2.9$ Now we have the logarithm isolated, which is exactly what we wanted!

STEP 4

Remember that logarithms and exponents are inverse operations, just like addition and subtraction, or multiplication and division.
The equation $\log _{8} x = 2.9$ means "8 raised to the power of $2.9$ equals $x$ ".
We can write this in exponential form as: $x = 8^{2.9}$ This looks much easier to solve now, doesn't it?

STEP 5

Let's calculate the value of $8^{2.9}$ .
Using a calculator, we find: $x = 8^{2.9} \approx 364.039$ Remember, we need to round to three decimal places, as requested!

STEP 6

The solution is $x \approx 364.039$ .
So we choose A.
The solution is $x \approx$ $364.039$ .

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Studdy Solution

STEP 1

What is this asking? We need to find the value of $x$ that satisfies the logarithmic equation $2 \log _{8} x = 5.8$ . Watch out! Remember the properties of logarithms, especially how to change the base and convert between logarithmic and exponential forms!

STEP 2

1. Isolate the logarithm
2. Convert to exponential form
3. Calculate the result

STEP 3

STEP 4

STEP 5

Let's calculate the value of $8^{2.9}$ .
Using a calculator, we find: $x = 8^{2.9} \approx 364.039$ Remember, we need to round to three decimal places, as requested!

STEP 6

The solution is $x \approx 364.039$ .
So we choose A.
The solution is $x \approx$ $364.039$ .

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Studdy Solution

STEP 1

What is this asking? We need to find the value of xxx that satisfies the logarithmic equation 2log⁡8x=5.82 \log _{8} x = 5.82log8​x=5.8. Watch out! Remember the properties of logarithms, especially how to change the base and convert between logarithmic and exponential forms!

STEP 2

1. Isolate the logarithm2. Convert to exponential form3. Calculate the result

STEP 3

STEP 4

STEP 5

Let's **calculate** the value of 82.98^{2.9}82.9.Using a calculator, we find: x=82.9≈364.039 x = 8^{2.9} \approx 364.039 x=82.9≈364.039 Remember, we need to round to three decimal places, as requested!

STEP 6

The solution is x≈364.039x \approx 364.039x≈364.039.So we choose **A**.The solution is x≈x \approxx≈ 364.039364.039364.039.

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What is this asking? We need to find the value of $x$ that satisfies the logarithmic equation $2 \log _{8} x = 5.8$ . Watch out! Remember the properties of logarithms, especially how to change the base and convert between logarithmic and exponential forms!

1. Isolate the logarithm
2. Convert to exponential form
3. Calculate the result

Let's calculate the value of $8^{2.9}$ .
Using a calculator, we find: $x = 8^{2.9} \approx 364.039$ Remember, we need to round to three decimal places, as requested!

The solution is $x \approx 364.039$ .
So we choose A.
The solution is $x \approx$ $364.039$ .