Answer:
(a) Taking the natural logarithm of both sides of the equation, we get:
ln(e^0.8x) = ln(7)
Using the property of logarithms that ln(e^y) = y for any y, we can simplify the left-hand side to:
0.8x = ln(7)
Dividing both sides by 0.8, we obtain the exact solution:
x = ln(7)/0.8 ≈ 1.52816
(b) Using a calculator to approximate the solution to six decimal places, we get:
x ≈ 1.528157
Explanation:
To solve the exponential equation e^0.8x = 7, we can use logarithms to isolate x. Specifically, we can take the natural logarithm of both sides of the equation, because the natural logarithm is the inverse function of the exponential function e^x. This gives us:
ln(e^0.8x) = ln(7)
Using the property of logarithms that ln(a^b) = b ln(a) for any base a and exponent b, we can simplify the left-hand side to:
0.8x ln(e) = ln(7)
Since ln(e) = 1, this further simplifies to:
0.8x = ln(7)
Finally, we can divide both sides by 0.8 to get the exact solution for x:
x = ln(7)/0.8 ≈ 1.52816
To find an approximate solution to the equation rounded to six decimal places, we can use a calculator to evaluate ln(7)/0.8, which gives us:
x ≈ 1.528157
This is an approximation of the exact solution that is accurate to six decimal places.