Natural Logs 24-25

Front

What is the natural logarithm?

Back

The natural logarithm, denoted as ln(x), is the logarithm to the base e, where e is approximately equal to 2.71828. It is the inverse function of the exponential function e^x.

Front

What is the relationship between natural logs and exponential functions?

Back

Natural logarithms and exponential functions are inverses of each other. If y = ln(x), then x = e^y.

Front

How do you solve an equation involving natural logs?

Back

To solve an equation involving natural logs, isolate the logarithmic term, exponentiate both sides to eliminate the log, and then solve for the variable.

Front

What is the value of e?

Back

The value of e is approximately 2.71828, and it is the base of natural logarithms.

Front

How do you convert a natural logarithm to exponential form?

Back

To convert ln(a) = b to exponential form, rewrite it as e^b = a.

Front

What is the derivative of the natural logarithm function?

Back

The derivative of ln(x) is 1/x for x > 0.

Front

What is the integral of the natural logarithm function?

Back

The integral of ln(x) is x(ln(x) - 1) + C, where C is the constant of integration.

Front

How do you solve the equation 5e^(3x - 2) + 15 = 35?

Back

First, isolate the exponential term: 5e^(3x - 2) = 20. Then divide by 5: e^(3x - 2) = 4. Take the natural log of both sides: 3x - 2 = ln(4). Solve for x: x = (ln(4) + 2)/3, which approximates to 1.13.

Front

What is the solution to the equation 3e^(2x) + 10 = 25?

Back

Isolate the exponential term: 3e^(2x) = 15. Divide by 3: e^(2x) = 5. Take the natural log: 2x = ln(5). Solve for x: x = (ln(5))/2, which approximates to 0.805.

Front

How do you solve e^(x + 6) = 12 for x?

Back

Take the natural log of both sides: x + 6 = ln(12). Solve for x: x = ln(12) - 6.