2.6 Warm Up: Base e and Natural Logarithms

Front

What is the natural logarithm of a number?

Back

The natural logarithm of a number is the logarithm to the base e, where e is approximately equal to 2.71828. It is denoted as ln(x).

Front

What is the value of e raised to the power of 0?

Back

e^0 = 1.

Front

How do you solve the equation 2e^x - 3 = 1 for x?

Back

First, isolate e^x: 2e^x = 4, then e^x = 2. Finally, take the natural logarithm: ln(2) = x.

Front

What is the relationship between exponential and logarithmic functions?

Back

Exponential functions and logarithmic functions are inverses of each other. If y = e^x, then x = ln(y).

Front

How do you approximate the solution to ln(x + 3) = 5?

Back

To approximate, exponentiate both sides: x + 3 = e^5, then solve for x: x = e^5 - 3.

Front

What is the equivalent logarithmic form of e^x = 8?

Back

The equivalent logarithmic form is ln(8) = x.

Front

How do you solve the equation e^{3x} - 5 = 32 for x?

Back

First, isolate e^{3x}: e^{3x} = 37. Then take the natural logarithm: 3x = ln(37), and finally x = ln(37)/3.

Front

What is the exact solution to the equation 2e^{5x} = 24?

Back

The exact solution is x = ln(12)/5.

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 value of ln(1)?

Back

ln(1) = 0.