14.1: Algebraic Properties of exp and log
- Page ID
- 49038
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)We recall some of the main properties of the exponential function:
\[\boxed{\begin{matrix} \quad b^{x + y} = b^x\cdot b^y\quad \\ b^{x-y} = \dfrac{b^x}{b^y} \\ (b^x)^n = b^{nx} \end{matrix}} \nonumber \]
Writing the above properties in terms of \(f(x)=b^x\) we have \(f(x+y)=f(x)f(y)\), \(f(x-y)=f(x)/f(y)\), and \(f(nx)=f(x)^n\).
Due to the fact that the logarithm is the inverse function of the exponential, we have corresponding properties whose proof follows:
The logarithm behaves well with respect to products, quotients, and exponentiation. Indeed, for all positive real numbers \(0<b\neq 1\), \(x>0\), \(y>0\), and real numbers \(n\), we have:
\[\boxed{\begin{matrix} \quad\log_b(x\cdot y) = \log_b(x)+\log_b(y)\quad \\ \log_b(\dfrac{x}{y}) = \log_b(x)-\log_b(y) \\ \log_b(x^n) = n\cdot \log_b(x) \end{matrix}} \nonumber \]
In terms of the logarithmic function \(g(x)=\log_b(x)\), the properties in the table above can be written: \(g(xy)=g(x)+g(y)\), \(g(x/y)=g(x)-g(y)\), and \(g(x^n)=n\cdot g(x)\).
Furthermore, for another positive real number \(0<a\neq 1\), we have the change of base formula:
\[\label{EQU:log-change-base} \boxed{\quad\log_b(x)=\dfrac{\log_a(x)}{\log_a(b)}\quad} \]
In particular, we have the formulas from equation 13.2.2 when taking the base \(a=10\) and \(a=e\):
\[\log_b(x) = \dfrac{\log(x)}{\log(b)}\quad\quad\text{and}\quad\quad \log_b(x) = \dfrac{\ln(x)}{\ln(b)} \nonumber \]
- Proof
-
We start with the first formula \(\log_b(x\cdot y) = \log_b(x)+\log_b(y)\). If we call \(u=\log_b(x)\) and \(v=\log_b(y)\), then the equivalent exponential formulas are \(b^u=x\) and \(b^v=y\). With this, we have
\[x\cdot y = b^u\cdot b^v =b^{u+v} \nonumber \]
Rewriting this in logarithmic form, we obtain
\[\log_b(x\cdot y)=u+v=\log_b(x)+\log_b(y) \nonumber \]
This is what we needed to show.
Next, we prove the formula \(\log_b\left(\dfrac{x}{y}\right) = \log_b(x)-\log_b(y)\). We abbreviate \(u=\log_b(x)\) and \(v=\log_b(y)\) as before, and their exponential forms are \(b^u=x\) and \(b^v=y\). Therefore, we have
\[\dfrac{x}{y} = \dfrac{b^u}{b^v} =b^{u-v} \nonumber \]
Rewriting this again in logarithmic form, we obtain the desired result.
\[\log_b\Big(\dfrac{x}{y}\Big)=u-v=\log_b(x)-\log_b(y) \nonumber \]
For the third formula, \(\log_b(x^n) = n\cdot \log_b(x)\), we write \(u=\log_b(x)\), that is in exponential form \(b^u=x\). Then:
\[x^n=(b^u)^n=b^{n\cdot u} \quad \implies \quad \log_b(x^n)= n\cdot u =n\cdot \log_b(x) \nonumber \]
For the last formula \(\ref{EQU:log-change-base}\), we write \(u=\log_b(x)\), that is, \(b^u=x\). Applying the logarithm with base \(a\) to \(b^u=x\) gives \(\log_a(b^u)=\log_a(x)\). As we have just shown before, \(\log_a(b^u)=u\cdot \log_a(b)\). Combining these identities with the initial definition \(u=\log_b(x)\), we obtain
\[\log_a(x)= \log_a(b^u)= u\cdot \log_a(b) =\log_b(x)\cdot \log_a(b) \nonumber \]
Dividing both sides by \(\log_a(b)\) gives the result \(\dfrac{\log_a(x)}{\log_a(b)}=\log_b(x)\).
Combine the terms using the properties of logarithms so as to write as one logarithm.
- \(\dfrac{1}{2}\ln(x)+\ln(y)\)
- \(\dfrac{2}{3}(\log(x^2y)-\log(xy^2))\)
- \(2\ln(x)-\dfrac{1}{3}\ln(y)-\dfrac{7}{5}\ln(z)\)
- \(5+\log_2(a^2-b^2)-\log_2(a+b)\)
Solution
Recall that a fractional exponent can also be rewritten with an \(n\)th root.
\[\boxed{x^{\frac 1 2} = \sqrt x} \quad\text{ and }\quad \boxed{x^{\frac 1 n} = \sqrt[n]x} \quad\implies \quad x^{\frac p q}=(x^p)^{\frac{1} {q}}=\sqrt[q]{x^p} \nonumber \]
We apply the rules from proposition Logarithm Properties.
- \(\dfrac{1}{2}\ln(x)+\ln(y)= \ln(x^{\frac 1 2})+\ln(y)=\ln(x^{\frac 1 2} y)=\ln(\sqrt{x}\cdot y)\)
- \(\dfrac{2}{3}(\log(x^2y)-\log(xy^2))=\dfrac{2}{3}\left(\log\left(\dfrac{x^2y}{xy^2}\right)\right)=\dfrac{2}{3}\left(\log\left(\dfrac{x}{y}\right)\right)\) &\(=\log\left(\left(\dfrac{x}{y}\right)^{\frac{2}{3}}\right)=\log\left(\sqrt[3]{\dfrac {x^2} {y^2}}\,\,\right)\)
- \(2\ln(x)-\dfrac{1}{3}\ln(y)-\dfrac{7}{5}\ln(z)=\ln(x^2)-\ln(\sqrt[3]{y})-\ln(\sqrt[5]{z^7})=\ln \left(\dfrac{x^2}{\sqrt[3]{y}\cdot \sqrt[5]{z^7}}\right)\)
- \(5+\log_2(a^2-b^2)-\log_2(a+b)=\log_2(2^5)+\log_2(a^2-b^2)-\log_2(a+b)\) &\(=\log_2\left(\dfrac{2^5\cdot (a^2-b^2)}{a+b}\right)=\log_2\left(\dfrac{32\cdot (a+b)(a-b)}{a+b}\right)=\log_2(32\cdot(a-b))\)
Write the expressions in terms of elementary logarithms \(u=\log_b(x)\), \(v=\log_b(y)\), and, in part (c), also \(w=\log_b(z)\). Assume that \(x,y,z>0\).
- \(\ln( \sqrt{x^5}\cdot y^2)\)
- \(\log \left(\sqrt{\sqrt{x} \cdot y^{3}}\right)\)
- \(\log _{2}\left(\sqrt[3]{\dfrac{x^{2}}{y \sqrt{z}}}\right)\)
Solution
In a first step, we rewrite the expression with fractional exponents, and then apply the rules from proposition Logarithm Properties.
- \(\begin{aligned}
\ln \left(\sqrt{x^{5}} \cdot y^{2}\right)&=\ln \left(x^{\frac{5}{2}} \cdot y^{2}\right)\\
&=\ln \left(x^{\frac{5}{2}}\right)+\ln \left(y^{2}\right) \\
&=\dfrac{5}{2} \ln (x)+2 \ln (y)\\
&=\dfrac{5}{2} u+2 v
\end{aligned}\)
- \(\begin{aligned}
\log \left(\sqrt{\sqrt{x} \cdot y^{3}}\right)&=\log \left(\left(x^{\frac{1}{2}} y^{3}\right)^{\frac{1}{2}}\right)\\
&=\dfrac{1}{2} \log \left(x^{\frac{1}{2}} y^{3}\right) \\
&=\dfrac{1}{2}\left(\log \left(x^{\frac{1}{2}}\right)+\log \left(y^{3}\right)\right)\\
&=\dfrac{1}{2}\left(\dfrac{1}{2} \log (x)+3 \log (y)\right) \\
&=\dfrac{1}{4} \log (x)+\dfrac{3}{2} \log (y)\\
&=\dfrac{1}{4} u+\dfrac{3}{2} v
\end{aligned}\)
- \(\begin{aligned}\log _{2}\left(\sqrt[3]{\dfrac{x^{2}}{y \sqrt{z}}}\right) &=\log _{2}\left(\left(\dfrac{x^{2}}{y \cdot z^{\frac{1}{2}}}\right)^{\frac{1}{3}}\right)=\dfrac{1}{3} \log _{2}\left(\dfrac{x^{2}}{y \cdot z^{\frac{1}{2}}}\right) \\ &=\dfrac{1}{3}\left(\log _{2}\left(x^{2}\right)-\log _{2}(y)-\log _{2}\left(z^{\frac{1}{2}}\right)\right) \\ &=\dfrac{1}{3}\left(2 \log _{2}(x)-\log _{2}(y)-\dfrac{1}{2} \log _{2}(z)\right) \\ &=\dfrac{2}{3} \log _{2}(x)-\dfrac{1}{3} \log _{2}(y)-\dfrac{1}{6} \log _{2}(z) \\ &=\dfrac{2}{3} u-\dfrac{1}{3} v-\dfrac{1}{6} w \end{aligned}\)