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. Functions4h 53m

Introduction to Functions
16m

Piecewise Functions
10m

Properties of Functions
9m

Common Functions
1h 8m

Transformations
5m

Combining Functions
27m

Exponent Rules
32m

Exponential Functions
28m

Logarithmic Functions
24m

Properties of Logarithms
34m

Exponential & Logarithmic Equations
35m

1. Limits and Continuity2h 2m

Introduction to Limits
41m

Finding Limits Algebraically
40m

Continuity
40m

2. Intro to Derivatives1h 33m

Tangent Lines and Derivatives
30m

Derivatives as Functions
14m

Basic Graphing of the Derivative
23m

Differentiability
26m

3. Techniques of Differentiation2h 18m

Basic Rules of Differentiation
43m

Higher Order Derivatives
12m

Product and Quotient Rules
55m

The Chain Rule
27m

4. Derivatives of Exponential & Logarithmic Functions1h 16m

Derivatives of Exponential & Logarithmic Functions
1h 16m

5. Applications of Derivatives2h 19m

Implicit Differentiation
27m

Logarithmic Differentiation
17m

Related Rates
59m

Differentials
21m

Linearization
13m

6. Graphical Applications of Derivatives6h 0m

Intro to Extrema
21m

The First Derivative Test
1h 20m

Concavity
1h 0m

The Second Derivative Test
22m

Curve Sketching
44m

Finding Global Extrema
29m

Applied Optimization
1h 41m

7. Antiderivatives & Indefinite Integrals48m

Antiderivatives
17m

Indefinite Integrals
31m

8. Definite Integrals4h 30m

Estimating Area with Finite Sums
42m

Riemann Sums
1h 16m

Introduction to Definite Integrals
20m

Fundamental Theorem of Calculus
25m

Average Value of a Function
17m

Substitution for Indefinite Integrals
55m

Substitution for Definite Integrals
32m

9. Graphical Applications of Integrals1h 43m

Area Between Curves
1h 19m

Introduction to Volume & Disk Method
23m

10. Integrals of Inverse, Exponential, & Logarithmic Functions21m

Integrals of Exponential Functions
14m

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7m

11. Techniques of Integration2h 7m

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1h 37m

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29m

12. Trigonometric Functions6h 54m

Angles
29m

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1h 8m

Solving Right Triangles
23m

Trigonometric Functions on the Unit Circle
1h 19m

Graphs of Sine & Cosine
46m

Graphs of Other Trigonometric Functions
32m

Trigonometric Identities
52m

Derivatives of Trig Functions
42m

Integrals of Basic Trig Functions
28m

Integrals of Other Trig Functions
10m

13: Intro to Differential Equations2h 23m

Basics of Differential Equations
43m

Slope Fields
19m

Separable Differential Equations
1h 21m

14. Sequences & Series2h 8m

Sequences
1h 12m

Review of Factorials
11m

Series
44m

15. Power Series2h 19m

Power Series & Taylor Series
2h 19m

16. Probability & Calculus45m

Continuous Probability Models
45m

0. Functions

Exponential & Logarithmic Equations

Business Calculus

0. Functions

Exponential & Logarithmic Equations

Problema precedenteProssimo problema

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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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Problema precedenteProssimo problema

Guida verificata passo dopo passo

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} $.