Exponents and logarithms Contents Properties Each row shows the same property in exponential and logarithmic forms.
Let X = b x X = b^x X = b x Y = b y Y = b^y Y = b y b > 0 b > 0 b > 0
exponents logarithms b x = X b^x = X b x = X x = log b X x = \log_bX x = log b X b 0 = 1 b^0 = 1 b 0 = 1 log b ( 1 ) = 0 \log_b(1) = 0 log b ( 1 ) = 0 b log b x = x \displaystyle b^{\log_bx} = x b l o g b x = x log b ( b x ) = x \log_b(b^x) = x log b ( b x ) = x b x b y = b x + y b^x b^y=b^{x+y} b x b y = b x + y log b X + log b Y = log b ( X Y ) \displaystyle \log_b{X} + \log_bY=\log_b(XY) log b X + log b Y = log b ( X Y ) b 1 n = b n \displaystyle b^{\frac1n} = \sqrt[n]{b} b n 1 = n b b − x = 1 b x \displaystyle b^{-x} = \frac{1}{b^x} b − x = b x 1 log b ( 1 X ) = − log b ( X ) \displaystyle \log_b\left(\frac{1}X\right) = -\log_b(X) log b ( X 1 ) = − log b ( X ) ( b x ) y = b x y = Z (b^x)^y = b^{xy} = Z ( b x ) y = b x y = Z log b X y = y log b X log X Z = log b Z log b X log b x Z = 1 x log b Z \displaystyle \log_b{X^y} = y\log_bX \\ \log_X Z = \frac{\log_bZ}{\log_bX} \\ \log_{b^x}Z =\frac1{x}\log_b{Z} log b X y = y log b X log X Z = log b X log b Z log b x Z = x 1 log b Z
These properties does not assume any relationships between the variables, rather they are used this way to help highlight the connections between the exponential and the equivalent logarithmic forms.
Strategies Take natural log (ln \ln ln Convert to the same base. Let y = log x y = \log x y = log x Example: Solve log 4 ( x 2 ) + 4 = 2 ( log 4 x ) 2 \log_4(x^2) + 4 = 2\left(\log_4 x\right)^2 log 4 ( x 2 ) + 4 = 2 ( log 4 x ) 2
log 4 ( x 2 ) + 4 = 2 ( log 4 x ) 2 2 log 4 x + 4 = 2 ( log 4 x ) 2 0 = 2 ( log 4 x ) 2 − 2 log 4 x − 4 0 = ( log 4 x ) 2 − log 4 x − 2 0 = ( log 4 x − 2 ) ( log 4 x + 1 ) log 4 x = 2 , − 1 x = 4 2 , 4 − 1 x = 16 , 1 4 ■ \begin{align*} \log_4(x^2) + 4 &= 2\left(\log_4 x\right)^2\\
2\log_4 x + 4 &= 2\left(\log_4 x\right)^2 \\
0 &= 2\left(\log_4 x\right)^2 - 2\log_4 x - 4 \\
0 &= \left(\log_4 x\right)^2 - \log_4 x - 2 \\
0 &= \left(\log_4 x - 2\right)\left(\log_4 x + 1\right) \\
\log_4 x &= 2, -1 \\
x &= 4^2, 4^{-1} \\
x &= 16, \frac{1}{4} \qed
\end{align*} log 4 ( x 2 ) + 4 2 log 4 x + 4 0 0 0 log 4 x x x = 2 ( log 4 x ) 2 = 2 ( log 4 x ) 2 = 2 ( log 4 x ) 2 − 2 log 4 x − 4 = ( log 4 x ) 2 − log 4 x − 2 = ( log 4 x − 2 ) ( log 4 x + 1 ) = 2 , − 1 = 4 2 , 4 − 1 = 16 , 4 1 ■ Order of operations a b c = a ( b c ) ≠ ( a b ) c a^{b^c} = a^{\left(b^c\right)} \neq \left(a^b\right)^c a b c = a ( b c ) = ( a b ) c Exponents are evaluated top to bottom.
Notes The n n n 3 3 3 − 3 -3 − 3 9 9 9 9 = 3 \sqrt{9} = 3 9 = 3 9 ≠ − 3 \sqrt{9} \neq -3 9 = − 3 x n \sqrt[n]{x} n x fractional exponents of complex numbers (HL) and fundamendal theorem of algebra (HL) .