Answer: To write an equivalent logarithmic equation to e^8.2 = 10x, we use the fact that log base 10 is the inverse of exponential base 10. Therefore, we have:
log(10)(e^8.2) = log(10)(10x)
Using the property of logarithms that says log base a of a^b is equal to b, we get:
8.2 = log(10)(10x)
Using the fact that log base 10 is commonly written as just "log," we can simplify this to:
8.2 = log(10x)
This is the equivalent logarithmic equation.
To write an equivalent exponential equation to In0.0002 = x, we use the fact that In is the inverse of e^x. Therefore, we have:
e^(In0.0002) = e^x
Using the property of logarithms that says e^ln(a) = a, we get:
0.0002 = e^x
This is the equivalent exponential equation.
To write an equivalent exponential equation to In(4x) = 9.6, we use the fact that In is the inverse of e^x. Therefore, we have:
e^(In(4x)) = e^9.6
Using the property of logarithms that says e^ln(a) = a, we get:
4x = e^9.6
Dividing both sides by 4, we get:
x = (1/4)e^9.6
This is the equivalent exponential equation.
Explanation: