Exponents and logarithms

Properties
- Limitations about these formulas
Strategies
Order of operations
Notes

Properties

Each row shows the same property in exponential and logarithmic forms.

Let $X = b^x$ , $Y = b^y$ . For $b > 0,\, b \neq 1$ :

exponents	logarithms
$b^x = X$	$x = \log_bX$
$b^0 = 1$	$\log_b(1) = 0$
$\displaystyle b^{\log_bx} = x$	$\log_b(b^x) = x$
$b^x b^y=b^{x+y}$	$\displaystyle \log_b{X} + \log_bY=\log_b(XY)$
$\displaystyle b^{\frac1n} = \sqrt[n]{b}$
$\displaystyle b^{-x} = \frac{1}{b^x}$	$\displaystyle \log_b\left(\frac{1}X\right) = -\log_b(X)$
$(b^x)^y = b^{xy} = Z$	$\displaystyle \log_b{X^y} = y\log_bX \\ \log_X Z = \frac{\log_bZ}{\log_bX} \\ \log_{b^x}Z =\frac1{x}\log_b{Z}$

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. It may help with understanding if you prove the logarithmic properties using the exponential properties, or vice versa.

Proof: From the formulas available in the formula booklet, show that

\log_{b^x} Z= \frac1x \log_b Z, \,b > 0,\, b \neq 1

From change of base formula

\log_{b^x} Z = \frac{\log_b Z}{\log_b b^x} = \frac{\log_b Z}{x}\qed

Limitations about these formulas

The base $b$ in both exponential and logarithmic functions are positive. Rational exponents of negative bases are touched upon in complex numbers.

The formula $\log_b{X} + \log_bY=\log_b(XY)$ only applies for positive $X, Y$ . For instance under the real number system, $ln (-1)$ is not defined so we cannot write $\ln (-x)$ as $\ln x + \ln(-1)$ . However, under analytic continuation (very much beyond the scope of IB mathematics), logarithms of negative numbers (and other complex numbers) are defined in a way such that this property holds.

Strategies

Goal is either $\log_b x = k$ and use $x = b^k$ , or $b^x = m$ and use $x = \log_b m$ , where $k$ and $m$ are expressions that do not involve $x$ .
For exponential equations, take natural log ( $\ln$ ) of both sides; for logarithmic equations, convert to same base then raise the base to both sides.
Let $y = \log x$ or let $y = b^x$ . This works when the resultant equation in $y$ is linear or quadratic or another equation you could solve.

Example: Solve $\log_4(x^2) + 4 = 2\left(\log_4 x\right)^2$

\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*}

Order of operations

a^{b^c} = a^{\left(b^c\right)} \neq \left(a^b\right)^c

Exponents are evaluated top to bottom.

Notes

The $n$ th roots of a number are not unique. While both $3$ and $-3$ are both square roots of $9$ , $\sqrt{9} = 3$ but $\sqrt{9} \neq -3$ . $\sqrt[n]{x}$ typically refer to what’s known as the “principal root”, though in some cases it could refer to the negative real root, if there is no positive real root. For more information see fractional exponents of complex numbers (HL) and fundamendal theorem of algebra (HL).