The logarithm function, log_b(x), is the inverse function of the exponential function, b^x. Here b is called the base of the logarithm and for our purposes b can be any positive number other than 1. The logarithm function is only defined for positive inputs. The output is negative for input values between 0 and 1 and positive for input values greater than 1. In other words, the domain is (0, ∞) and the range is (–∞, ∞). For an input value of 1, log_b(1) = 0.

Just like applying any other mathematical function, we can apply the base-b logarithm function (called “taking the base-b log”) to a number or variable or algebraic expression. For example, if y = b^x, then we can take the base-b log of both sides to obtain log_b(y) = log_b(b^x), which simplifies to log_b(y) = x (since the log function and exponential functions are inverse functions).

Equivalently, if we start with the equation log_b(y) = x, we can exponentiate both sides using base b to obtain b^logb(y) = b^x, which simplifies to y = b^x (since the exponential function and log functions are inverse functions).

Calculators and computers can typically evaluate logarithms for two bases directly:

• log₁₀(x) with base 10 is the common log, typically written simply as log(x).
• log_e(x) with base e is the natural log, typically written simply as ln(x). Here the constant e is Euler’s number, which represents the limit of (1+1/k)^k as k gets very large. As a decimal, e is approximately 2.718282…

Exponential equations

Thus, exponential equations involving base 10 or base e are straightforward to solve:

• For the equation x = 10^y, find y if x = 40.
  • First, take the common log of both sides: log(x) = log(10^y) = y.
  • Next, evaluate y = log(40) using a calculator or computer: y = log(40) ≈ 1.6021 (to 4 decimal places).
  • Check: 10^1.6021 = 40.00.
• For the equation x = e^y, find y if x = 5.
  • First, take the natural log of both sides: ln(x) = log(e^y) = y.
  • Next, evaluate y = ln(5) using a calculator or computer: y = ln(5) ≈ 1.6094 (to 4 decimal places).
  • Check: e^1.6094 = 5.00. (On a calculator or computer, look for the button or function labeled e^x or exp(x).)

Logarithmic equations

Similarly, logarithmic equations involving base 10 or base e are straightforward to solve:

• For the equation x = log(y), find y if x = 2.
  • First, exponentiate both sides using base 10: 10^x = 10^log(y) = y.
  • Next, evaluate y = 10² = 100.
  • Check: log(100) = 2.
• For the equation x = ln(y), find y if x = 5.
  • First, exponentiate both sides using base e: e^x = e^ln(y) = y.
  • Next, evaluate y = e⁵ using a calculator or computer: y = e⁵ ≈ 148.41 (to 2 decimal places).
  • Check: ln(148.41) = 5.00.