Logarithms to base 10 are called common logarithms. We often write “log10” as “log” or “lg”. Common logarithms can be evaluated using a scientific calculator.
Recall that by the definition of logarithm.
log Y = X ↔ Y = 10X
Besides base 10, another important base is e. Log to base e are called natural logarithms. “loge” are often abbreviated as “ln”. Natural logarithms can also be evaluated using a scientific calculator.
ln Y = X ↔ Y = eX
Using a calculator, we can use common and natural logarithms to solve equations of the form ax = b, especially when b cannot be expressed as an.
Solve the equations
a) 6x + 2 = 21
b) e2x = 9
a) 6x + 2 = 21
log 6x + 2 = log 21
(x + 2) log 6 = log 21
b) e3x = 9
ln e3x = ln 9
3x ln e = ln 9
3x = ln 9
Express 3x(22x) = 7(5x) in the form ax = b. Hence, find x.
Solution:Since 3x(22x) = 3x(22)x = (3 × 4)x = 12x
the equation becomes
12x = 7(5x)
Common and Natural Logarithms
We can use many bases for a logarithm, but the bases most typically used are the bases of the common logarithm and the natural logarithm. The common logarithm has base 10, and is represented on the calculator as log(x). The natural logarithm has base e, a famous irrational number, and is represented on the calculator by ln(x). The natural and common logarithm can be found throughout Algebra and Calculus.
Defines common log, log x, and natural log, ln x, and works through examples and problems using a calculator.
Rotate to landscape screen format on a mobile phone or small tablet to use the Mathway widget, a free math problem solver that answers your questions with step-by-step explanations.
We welcome your feedback, comments and questions about this site or page. Please submit your feedback or enquiries via our Feedback page.