Solve: 0.3*10^(3x) = 32
(again says)
a. With an exact answer.
b. Decimal form with one decimal accuracy.
c. Decimal form with three values of accuracy.
why all that work?
0.3*10^(3x) = 32
10^(3x) = 32/.3
3x = log(32/.3)
x = 1/3 log(32/.3)
0.3*10^(3x)=32
3×10^(3x-1)=32
10^(3x-1)=32/3
(3x-1)log10=(log32-log3)
3x-1=(log32-log3)
3x-1=1.02803
3x=1.02803+1=2.02803
X=(2.02803)/3
I don't know why, it's part of the question. :(
I've honestly not enjoyed this school here lol. I'm hoping to get to another one. They don't help at all (they toss the book at you and theres no actual classes) so if you don't understand the book your sol. Reason why I ask here is mostly to try to grasp a couple since I'm doing 100-200 questions a week or around.
0.3*10^(3x) = 32
10^(3x) = 106.67
3x*Log10 = Log 106.67
3x = 2.028
X = 0.676.
To solve the equation 0.3 * 10^(3x) = 32, we can use logarithms. Here's how you can solve it step by step:
Step 1: Start by taking the logarithm of both sides of the equation. The most common logarithm used is the natural logarithm (ln), but you can also use the common logarithm (log) if you prefer.
ln(0.3 * 10^(3x)) = ln(32)
Step 2: Apply the logarithm properties to simplify the equation. We can use the multiplication rule, which states that ln(a * b) is equal to ln(a) + ln(b), and the exponent rule, which states that ln(a^b) is equal to b * ln(a).
ln(0.3) + ln(10^(3x)) = ln(32)
Step 3: Simplify further. We know that ln(10) = 2.303, so ln(10^(3x)) becomes (3x * 2.303).
ln(0.3) + (3x * 2.303) = ln(32)
Step 4: Use the properties of logarithms to separate the terms with x on one side of the equation. In this case, we'll move ln(0.3) to the other side by subtracting it from both sides.
(3x * 2.303) = ln(32) - ln(0.3)
Step 5: Evaluate the natural logarithm expressions on the right side using a scientific calculator or an online calculator.
(3x * 2.303) = 3.465
Step 6: Divide both sides of the equation by 2.303 to isolate the x term.
3x = 3.465 / 2.303
Step 7: Solve for x by dividing both sides by 3.
x = (3.465 / 2.303) / 3
Now, let's go through each part of the question:
a. With an exact answer:
To find the exact answer, calculate the value of x using the equation we derived in step 7.
x = (3.465 / 2.303) / 3 = 0.501
Therefore, the exact value of x is 0.501.
b. Decimal form with one decimal accuracy:
Using a calculator, evaluate the expression to one decimal digit accuracy:
x ≈ 0.5
So, the decimal form of x with one decimal accuracy is 0.5.
c. Decimal form with three decimal digits accuracy:
Again, using a calculator, evaluate the expression with three decimal digits accuracy:
x ≈ 0.501
Thus, the decimal form of x with three decimal digits accuracy is 0.501.