Natural Logs and e

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 value of e?

Back

The value of e is approximately 2.71828. It is a mathematical constant that is the base of the natural logarithm.

Front

How do you solve for x in the equation ln(x - 3) = 5?

Back

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

Front

What is the relationship between natural logarithms and exponents?

Back

Natural logarithms are the inverse of exponential functions. If y = e^x, then x = ln(y).

Front

How do you solve for x in the equation e^(5x) = 10?

Back

Take the natural logarithm of both sides: 5x = ln(10), then x = ln(10)/5.

Front

What is the definition of log base 3?

Back

Log base 3, denoted as log_3(x), is the power to which the base 3 must be raised to obtain the number x.

Front

How do you convert an exponential equation to logarithmic form?

Back

The exponential equation a^b = c can be converted to logarithmic form as log_a(c) = b.

Front

What is the solution to the equation 3^(2x - 6) = 81?

Back

To solve, rewrite 81 as 3^4: 2x - 6 = 4, then x = 5.

Front

How do you isolate x in the equation 4e^(x-2) = 28?

Back

Divide both sides by 4: e^(x-2) = 7, then take the natural logarithm: x - 2 = ln(7), so x = ln(7) + 2.

Front

What is the solution to the equation e^(x + 6) = 12?

Back

Take the natural logarithm of both sides: x + 6 = ln(12), then x = ln(12) - 6.