| Definition |
\(y=\log_b x\iff x=b^y\) |
The base is the same
\(b\gt 0;b\neq 1\) |
| Comparison |
\(\log_b x=\log_b y\iff x=y\) |
|
| Product rule |
\(\log_b(xy)=\log_b x+\log_b y\) |
The arguments, not the logarithms, are multiplied |
| Quotient rule |
\(\log_b\left(\dfrac{x}{y}\right)=\log_b x-\log_b y\) |
The arguments, not the logarithms, are divided |
| Power rule |
\(\log_b x^n=n\log_b x\) |
|
| Change-of-base rule |
\(\log_b x=\dfrac{\log_a x}{\log_a b}\) |
The logarithms, not the arguments, are divided.
What is above (\(x\)) remains above,
what is below (\(b\)) remains below |
| Inverse-function rules |
\(b^{\log_b x}=x\) |
|
| \(\log_b b^x=x\) |
|
| Special bases |
\(\log x=\log_{10} x\) |
Common logarithm is base 10 |
| \(\ln x=\log_e x\) |
Natural logarithm
\(e\) is ‘Euler's number’
\(e\approx 2.718281828\) |
| Special arguments |
\(\log_b b=1\) |
|
| \(\log_b 1=0\) |
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