The logarithm function

\log_2\space16=4\Leftrightarrow2^4=16

Common bases of logarithms

• Base 10: historically, logarithms were written down in ‘logarithm books’. These books showed all of their logarithms to base 10.
• Base e: natural logarithms. Euler’s number is the ‘natural base’ of logarithms.

Natural logarithm function

\ln(x)=y\Leftrightarrow \log_e(x)=y\Leftrightarrow e^{y}=x

Essentially, \ln(x) is just another way of writing \log_e(x). We use e all the time in logarithms, so it is easier to write \ln(x) instead of \log_e(x).

Logarithm function constraints

• In the function \log_aa^x:
  • a>0
  • a\neq1
  • x>0

Example: solve for x: 3\ln(2+x)=6

• Divide both sides by 3:
  • \ln(2+x)=2
• Rewrite the logarithm as a power of e:
  • e^2=2+x
• Subtract 2 from both sides:
  • e^2-2=x
• Therefore, the solution is:
  • x=e^2-2