Solving Exponential and Logarithmic Equations In Exercise, solve for x.
500(1.075)120x = 100.000
The correct Answer and Explanation is:
To solve the equation 500(1.075)120x=100,000500(1.075)^{120x} = 100,000500(1.075)120x=100,000, we can follow these steps:
Step 1: Isolate the Exponential Expression
First, we need to isolate the term with the exponential expression. To do this, divide both sides of the equation by 500:(1.075)120x=100,000500(1.075)^{120x} = \frac{100,000}{500}(1.075)120x=500100,000
Calculating the right side gives:(1.075)120x=200(1.075)^{120x} = 200(1.075)120x=200
Step 2: Take the Logarithm of Both Sides
Next, we apply the logarithm to both sides of the equation. You can use either natural logarithm (ln\lnln) or common logarithm (log\loglog). Here, we’ll use natural logarithm:ln((1.075)120x)=ln(200)\ln((1.075)^{120x}) = \ln(200)ln((1.075)120x)=ln(200)
Using the logarithmic identity ln(ab)=b⋅ln(a)\ln(a^b) = b \cdot \ln(a)ln(ab)=b⋅ln(a), we can simplify the left side:120x⋅ln(1.075)=ln(200)120x \cdot \ln(1.075) = \ln(200)120x⋅ln(1.075)=ln(200)
Step 3: Solve for xxx
Now, we can isolate xxx by dividing both sides by 120⋅ln(1.075)120 \cdot \ln(1.075)120⋅ln(1.075):x=ln(200)120⋅ln(1.075)x = \frac{\ln(200)}{120 \cdot \ln(1.075)}x=120⋅ln(1.075)ln(200)
Step 4: Calculate the Values
Next, we need to compute the values.
- Calculate ln(200)\ln(200)ln(200):ln(200)≈5.2983\ln(200) \approx 5.2983ln(200)≈5.2983
- Calculate ln(1.075)\ln(1.075)ln(1.075):ln(1.075)≈0.0723\ln(1.075) \approx 0.0723ln(1.075)≈0.0723
- Now plug these values into the formula for xxx:x≈5.2983120⋅0.0723x \approx \frac{5.2983}{120 \cdot 0.0723}x≈120⋅0.07235.2983 x≈5.29838.676≈0.609x \approx \frac{5.2983}{8.676} \approx 0.609x≈8.6765.2983≈0.609
Final Answer
Thus, the solution for xxx is approximately:0.609\boxed{0.609}0.609
Explanation
This solution involves isolating the exponential expression and applying logarithms, which is a standard technique for solving equations involving exponents. The use of logarithms allows us to turn the exponentiation into a product, which is easier to manipulate. Calculating the natural logarithm provides specific numerical values that can be used to solve for xxx. This process is widely applicable in exponential growth or decay problems in various fields, including finance and science, making it an essential skill in mathematics.