Answer :
For the first equation, x ≈ 8.04. For the second equation, x ≈ 1.92.
1. To solve for x, we need to isolate it on one side of the equation. We can do this by dividing both sides of the equation by 500:
10000/500 = (1.09)ˣ
20 = (1.09)ˣ
Now, we need to take the logarithm of both sides of the equation to eliminate the exponent. We can use either the natural logarithm (ln) or the common logarithm (log). In this case, let's use the natural logarithm:
ln(20) = ln((1.09)ˣ)
Using the property of logarithms that states ln(aᵇ) = b * ln(a), we can rewrite the equation as: ln(20) = x * ln(1.09)
Now, we can divide both sides of the equation by ln(1.09) to solve for x:
x = ln(20) / ln(1.09)
Using a calculator, we find that x ≈ 8.04 (rounded to two decimal places).
2. Moving on to the second equation: 60000 = 1200e²ˣ⁻¹
To solve for x, we will first divide both sides of the equation by 1200:
60000/1200 = e²ˣ⁻¹
50 = e²ˣ⁻¹
Next, we will take the natural logarithm of both sides of the equation:
ln(50) = ln(e²ˣ⁻¹)
Using the property of logarithms mentioned earlier, we can simplify the equation as follows: ln(50) = (2x-1) * ln(e)
Since ln(e) equals 1, we have: ln(50) = 2x - 1
Now, we will add 1 to both sides of the equation: ln(50) + 1 = 2x
Finally, we divide both sides of the equation by 2 to solve for x:
x = (ln(50) + 1) / 2
Using a calculator, we find that x ≈ 1.92 (rounded to two decimal places).
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The complete question is, " Solve to the nearest 100th: 10000 = 500(1.09)ˣ
Solve to the nearest 100th: 60000 = 1200e²ˣ⁻¹ "
