Solve the exponential equation. 2 ⋅ 1 0 3 x = 5000 2\cdot10^{3x}=5000 2 ⋅ 1 0 3 x = 5000

Here we're asked to solve the exponential equation, and we're given 2 times 10 to the power of 3x is equal to 5,000. Now taking a look at our steps starting with step 1, we first want to isolate our exponential expression. And here we have our exponential expression as 10 to the power of 3 x so we want to go ahead and get rid of that 2 and isolate that. So in order to get rid of that 2 I'm just going to divide both sides by 2 and that will cancel over here leaving me with just 10 to the power of 3 x And that is equal to 5,000 divided by 2, which is just going to be 2,500. Now we're done with step 1. We can go ahead and move on to step 2, where we're going to take our logs. Now since this is base 10, we want to go ahead and just take the log, the common log of both sides. So we're going to take log of 10 to the power of 3x, and that is equal to log of 25100. Remember, whatever you do to one side, you have to do to the other. So that's why we're taking the log of both sides. Now we want to go ahead and use our log rules to get x out of the exponent, moving on to that step 3. So this 3x, I can go ahead and pull to the front of that log. So this becomes 3x times log of 10, and that's equal to a log of 25100. Now I can go ahead and simplify this further because log of 10 is just 1 because this is really log base 10 of 10. So this is just 3x on this side because this cancels to a 1, and this is just 3 x is equal to log of 25100. Now moving on to step 4, we want to go ahead and solve for x, which the only thing we need to do in order to isolate x is divide both sides by 3. So my answer is log of 25 100 divided by 3, which if you're keeping it in log form, then you're completely done. But if we want to approximate this using a calculator, this is exactly what we're going to plug in, log of 25100 over 3. Now, when you do this, rounding to the nearest 100th place, you should get 1.13, and that is our final answer. If we were to plug this back into our original equation for x, we would end up with a true statement. Thanks for watching, and let me know if you have questions.

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0. Functions

Exponential & Logarithmic Equations

Business Calculus

0. Functions

Exponential & Logarithmic Equations

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객관식

Solve the exponential equation.
$2$\cdot$10^{3x}=5000$

$x=3.40$

$x=10.19$

$x=0.0001$

$x=1.13$

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Rewrite the given equation $ 2 \cdot 10^{3x} = 5000 $ by isolating the exponential term. Divide both sides of the equation by 2 to get $ 10^{3x} = 2500 $.

Take the logarithm of both sides to solve for the exponent. Use the common logarithm (base 10) for simplicity: $ \log(10^{3x}) = \log(2500) $.

Apply the logarithmic property $ \log(a^b) = b \cdot \log(a) $ to simplify the left-hand side: $ 3x \cdot \log(10) = \log(2500) $.

Recall that $ \log(10) = 1 $, so the equation simplifies to $ 3x = \log(2500) $.

Solve for $ x $ by dividing both sides of the equation by 3: $ x = \frac{\log(2500)}{3} $.