Common logarithm

Definition

A common logarithm is a logarithm with base 10, often written as $\log_{10}(x)$ or simply $\log(x)$. It is commonly used in scientific calculations and when dealing with exponential growth or decay.

5 Must Know Facts For Your Next Test

1. The common logarithm of 10 is 1, i.e., $\log(10) = 1$.
2. The common logarithm of 1 is 0, i.e., $\log(1) = 0$.
3. Common logarithms are the inverse functions of base-10 exponential functions.
4. The change of base formula allows conversion between natural logarithms and common logarithms: $\log_a(b) = \frac{\log(b)}{\log(a)}$.
5. $\log(xy) = \log(x) + \log(y)$ and $\log(\frac{x}{y}) = \log(x) - \log(y)$ are key properties used to simplify expressions.

Review Questions

• What is the value of $\log_{10}(100)$?
• How can you express $\ln(x)$ in terms of common logarithms using the change of base formula?
• Simplify the expression: $\log(2) + \log(5)$.

Related terms

Natural Logarithm: A logarithm with base e (approximately equal to 2.718), denoted as $\ln(x)$.

Exponential Function: A function of the form $f(x)=a^x$, where a is a positive constant.

Logarithmic Properties: $\text{Rules that apply to simplifying or manipulating log expressions such as } \text{product, quotient, and power rules}.$