Common and Natural Logarithms

A logarithm is the exponent to which a base must be raised to produce a given number. For example, if $$b^y = x$$ , then $$log_b(x) = y$$ .

The change of base formula states that $$log_b(a) = \frac{log_k(a)}{log_k(b)}$$ for any positive base $$k$$ .

The natural logarithm, denoted as $$ln(x)$$ , is the logarithm to the base $$e$$ , where $$e$$ is approximately equal to 2.718.

The common logarithm is the logarithm to the base 10, denoted as $$log(x)$$ .

Take the natural logarithm of both sides: $$x = ln(10)$$ , which is approximately 2.303.

The function shifts left by 4 units and up by 6 units.

$$log_5(75) = \frac{log(75)}{log(5)}$$ .

$$log_3(64) = \frac{log(64)}{log(3)}$$ .

The base of a logarithm is the number that is raised to a power to obtain a certain value. For example, in $$log_b(a)$$ , $$b$$ is the base.

The inverse of the logarithmic function is the exponential function. For example, if $$y = log_b(x)$$ , then $$x = b^y$$ .