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

flashcards

Question	Answer
The logarithm function	Logarithms are the inverse of exponentiation: \log_2{16}=4 is equivalent to 2^4=16.
Common bases of logarithms	Base 10: historically used in logarithm books. Base e: natural logarithms, where e is Euler’s number.
Natural logarithm function	\ln(x) = \log_e(x), so e^y = x is equivalent to \ln(x) = y, simplifying notation for base e.
Logarithm function constraints	In \log_a x, constraints are: a > 0, a \neq 1, x > 0.
How do you solve 3\ln(2+x)=6?	Divide by 3 to get \ln(2+x)=2, rewrite as e^2 = 2+x, then subtract 2: x = e^2 - 2.

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