9. Q9b
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Similar Question 1
<p>Simplify.</p> <ul> <li><code class='latex inline'>\displaystyle{3^{\log_327}+10^{\log_{10}1000}}</code></li> </ul>
Similar Question 2
<p>Solve for <code class='latex inline'>x</code>. Round your answers to two decimal places, if necessary.</p><p><code class='latex inline'>\log(1 000 000)=x</code></p>
Similar Question 3
<p>Evaluate without a calculator.</p><p><code class='latex inline'> \displaystyle 2^{ \log_2 5 - 1} </code></p>
Similar Questions
Learning Path
L1 Quick Intro to Factoring Trinomial with Leading a
L2 Introduction to Factoring ax^2+bx+c
L3 Factoring ax^2+bx+c, ex1
Now You Try
<p>The astronomer Johannes Kepler (1571~1630) determined that the time, <code class='latex inline'>D</code>, in days, for a planet to revolve around the Sun is related to the planet&#39;s average distance from the Sun, <code class='latex inline'>k</code>, in millions of kilometres. This relation is defined by the equation <code class='latex inline'>\displaystyle{\log D=\frac{3}{2}\log k-0.7}</code> Verify that Kepler&#39;s equation gives a good approximation of the time it takes for Earth to revolve around the Sun, if Earth is about 150 000 000 km from the Sun.</p>
<p>Use your knowledge of logarithms to solve each of the following equations for <code class='latex inline'>x</code>.</p><p><code class='latex inline'>\displaystyle{\log_{\frac{1}{4}}x=-2}</code></p>
<p>Evaluate each of the following:</p><p><code class='latex inline'> \displaystyle \log_327 </code></p>
<p>Evaluate each of the following:</p><p><code class='latex inline'> \displaystyle \log_24 </code></p>
<p> Evaluate the expression.</p><p>a) <code class='latex inline'> \displaystyle \log_{7}1 </code></p><p>b) <code class='latex inline'> \displaystyle \log_{4}{1024} </code></p>
<p>Use your knowledge of logarithms to solve each of the following equations for <code class='latex inline'>x</code>.</p><p> <code class='latex inline'>\displaystyle{\log_x27=3}</code></p>
<p>To evaluate a logarithm whose base is not 10 you can use the following relationship <code class='latex inline'>\displaystyle{\log_ab=\frac{\log b}{\log a}}</code> Use this to evaluate each of the following to four decimal places.</p><p> <code class='latex inline'>\log_55</code></p>
<p> Evaluate the following using the property <code class='latex inline'>a^{\log_a x} = x</code></p><p><code class='latex inline'> \displaystyle 2^{2\log_43 } </code></p>
<p>Use your knowledge of logarithms to solve each of the following equations for <code class='latex inline'>x</code>.</p><p><code class='latex inline'>\displaystyle{\log_4\frac{1}{64}=x}</code></p>
<p> Evaluate the expression.</p><p><code class='latex inline'> \displaystyle \log_{3}\sqrt{27} </code></p>
<p>Solve for <code class='latex inline'>x</code>. Round your answers to two decimal places, if necessary.</p><p><code class='latex inline'>\log(1 000 000)=x</code></p>
<p>Evaluate.</p><p><code class='latex inline'>\displaystyle{\log_mm^n}</code></p>
<p> Evaluate the following without a calculator</p><p><code class='latex inline'> \displaystyle \log_{0.5}16 </code></p>
<p>Simplify each expression.</p><p><code class='latex inline'>\displaystyle \ln e^{10} </code></p>
<p> Solve for <code class='latex inline'>x</code> in each of the following.</p><p><code class='latex inline'> \displaystyle \begin{array}{cccccc} &a) &3^x = 32 &b) &5^x = 66 \\ &c) &10^x = 157 &d) &1.03^x = 1.69 \\ &e) & 4.5^x = 600 &f) &\log 1.07^x = 2 \\ \end{array} </code></p>
<p>Determine the inverse of each relation.</p><p><code class='latex inline'>y=(0.5)^{x+2}</code></p>
<p> Evaluate the expression.</p><p><code class='latex inline'> \displaystyle \log_{7}\frac{1}{49} </code></p>
<p>Use your knowledge of logarithms to solve each of the following equations for <code class='latex inline'>x</code>.</p><p><code class='latex inline'>\displaystyle{\log_4x=1.5}</code></p>
<p> Evaluate the express without using a calculator.</p><p><code class='latex inline'> \displaystyle \log \log 10^{100} </code></p>
<p> Evaluate the expression.</p><p><code class='latex inline'> \displaystyle \log_{10}100 </code></p>
<p>Evaluate.</p><p><code class='latex inline'>\log_5\sqrt[3]{5}</code></p>
<p> Evaluate the expression.</p><p><code class='latex inline'> \displaystyle \log_{4}{4^{a + b}} </code></p>
<p>Solve for <code class='latex inline'>x</code>. Round your answers to two decimal places, if necessary.</p><p><code class='latex inline'>\log1=x</code></p>
<p>Evaluate each of the following:</p><p><code class='latex inline'> \displaystyle \log_3\sqrt{3} </code></p>
<p>Simplify.</p> <ul> <li><code class='latex inline'>\displaystyle{3^{\log_327}+10^{\log_{10}1000}}</code></li> </ul>
<p> Evaluate the following using the property <code class='latex inline'>\displaystyle \log_a a^x = x</code></p><p><code class='latex inline'> \displaystyle \log_{0.5}2 </code></p>
<p> Evaluate the expression.</p><p><code class='latex inline'> \displaystyle 2^{\log_{2}{13}} </code></p>
<p>To evaluate a logarithm whose base is not 10 you can use the following relationship <code class='latex inline'>\displaystyle{\log_ab=\frac{\log b}{\log a}}</code> Use this to evaluate each of the following to four decimal places.</p><p> <code class='latex inline'>\log_545</code></p>
<p>Solve for <code class='latex inline'>x</code>. Round your answers to two decimal places, if necessary.</p><p><code class='latex inline'>\displaystyle{\log\left(\frac{1}{10}\right)}</code></p>
<p>Evaluate.</p><p><code class='latex inline'>\displaystyle{4^{\log_4\frac{1}{16}}}</code></p>
<p>The doubling function <code class='latex inline'>\displaystyle{y=y_02^{\frac{t}{D}}}</code> can be used to model exponential growth when the doubling time is <code class='latex inline'>D</code>. The bacterium <code class='latex inline'>Escherichia</code> <code class='latex inline'>coli</code> has a doubling period of 0.32 h. A culture of <code class='latex inline'>E.</code> <code class='latex inline'>coli</code> starts with 100 bacteria.</p> <ul> <li>Graph your equation.</li> <li>Graph the inverse.</li> </ul>
<p>Express in logarithmic form.</p><p> <code class='latex inline'>8^0=1</code></p>
<p>Evaluate.</p><p><code class='latex inline'>\displaystyle{a^{\log_ab}}</code></p>
<p>Use your knowledge of logarithms to solve each of the following equations for <code class='latex inline'>x</code>.</p><p><code class='latex inline'>\displaystyle{\log_5x=\frac{1}{2}}</code></p>
<p> Evaluate the expression.</p><p><code class='latex inline'> \displaystyle \log_{2}\log_3(3^{128}) </code></p>
<p> Evaluate the following without a calculator</p><p><code class='latex inline'> \displaystyle \log_864 </code></p>
<p>Simplify.</p> <ul> <li><code class='latex inline'>\displaystyle{5^{\log_58}-3^{\log_35+\log_37}}</code></li> </ul>
<p> Change the following to logarithmic notation, <code class='latex inline'>y = \log_{a}x</code>.</p><p><code class='latex inline'> \displaystyle \begin{array}{ccccc} &a) & 3^5 = 243 &b) & 2^7 = 128 \\ &c) & 7^3 = 343 &d) & 11^4 = 14 641\\ \end{array} </code></p>
<p> Evaluate the following without a calculator</p><p><code class='latex inline'> \displaystyle \log_93 </code></p>
<p> Evaluate the following using the property <code class='latex inline'>a^{\log_a x} = x</code></p><p><code class='latex inline'> \displaystyle 5^{-\log_56} </code></p>
<p>To evaluate a logarithm whose base is not 10 you can use the following relationship <code class='latex inline'>\displaystyle{\log_ab=\frac{\log b}{\log a}}</code> Use this to evaluate each of the following to four decimal places.</p><p> <code class='latex inline'>\log_210</code></p>
<p>Consider the expression <code class='latex inline'>\log_5a</code>.</p> <ul> <li>For what values of <code class='latex inline'>a</code> will this expression yield positive numbers?</li> </ul>
<p>Evaluate without a calculator.</p><p><code class='latex inline'> \displaystyle 7^{0.5\log_749} </code></p>
<p>Graph the function and its inverse. State the domain, range, and asymptote of each. Determine the equation of the inverse.</p><p><code class='latex inline'>y=-2\log_53x</code></p>
<p>Evaluate.</p><p><code class='latex inline'>\displaystyle{\log_{\frac{1}{10}}1}</code></p>
<p> Evaluate the expression.</p><p><code class='latex inline'> \displaystyle 2^{\log_{2}{3}} + 1 </code></p>
<p> Evaluate the following using the property <code class='latex inline'>a^{\log_a x} = x</code></p><p><code class='latex inline'> \displaystyle 5^{\log_5 125} </code></p>
<p>Evaluate.</p><p><code class='latex inline'>\log_5125-\log_525</code></p>
<p>Consider the expression <code class='latex inline'>\log_5a</code>.</p> <ul> <li>For what values of <code class='latex inline'>a</code> will this expression yield negative numbers?</li> </ul>
<p> Change the following to logarithmic notation, <code class='latex inline'>x = a^y</code>.</p><p><code class='latex inline'> \begin{array}{ccccccc} &(a) & \log_{4}1024 = 5 &(b) &\log_{7}7 = 1 \\ &(c)& \log_{6}7776 = 5 &(d)& \log_{a}p = r \\ \end{array} </code></p>
<p>Evaluate without a calculator.</p><p><code class='latex inline'> \displaystyle 2^{ \log_2 5 - 1} </code></p>
<p>The function <code class='latex inline'>S(d)=93\log d+65</code> relates the speed of the wind, <code class='latex inline'>S</code>, in miles per hour, near the centre of a tornado to the distance that the tornado travels, <code class='latex inline'>d</code>, in miles.</p> <ul> <li>If a tornado has sustained winds of approximately 250 mph, estimate the distance it can travel.</li> </ul>
<p>Evaluate.</p><p><code class='latex inline'>\displaystyle{5^{\log_525}}</code></p>
<p> Evaluate the following using the property <code class='latex inline'>a^{\log_a x} = x</code></p><p><code class='latex inline'> \displaystyle 3^{\log_93} </code></p>
<p>Simplify each expression.</p><p><code class='latex inline'>\displaystyle \ln e^{3} </code></p>
<p> Evaluate the following using the property <code class='latex inline'>a^{\log_a x} = x</code></p><p><code class='latex inline'> \displaystyle 125^{-\log_56} </code></p>
<p> Evaluate the expression.</p><p><code class='latex inline'> \displaystyle \log \log 10^{100} </code></p>
<p>Evaluate without a calculator.</p><p><code class='latex inline'> \displaystyle 25^{\log_5 3} </code></p>
<p> Evaluate the following using the property <code class='latex inline'>a^{\log_a x} = x</code></p><p><code class='latex inline'> \displaystyle 25^{\log_5 125} </code></p>
<p> Find the value of </p><p><code class='latex inline'> \displaystyle (-2^{\log_{\sqrt{2}}\frac{1}{3}})\times(\log_{2}8) </code></p>
<p>Use your knowledge of logarithms to solve each of the following equations for <code class='latex inline'>x</code>.</p><p><code class='latex inline'>\displaystyle{\log_5x=3}</code></p>
<p>Graph the function and its inverse. State the domain, range, and asymptote of each. Determine the equation of the inverse.</p><p><code class='latex inline'>y=20(8)^x</code></p>
<p> Evaluate the following using the property <code class='latex inline'>a^{\log_a x} = x</code></p><p><code class='latex inline'> \displaystyle 27^{\log_93} </code></p>
<p>Evaluate.</p><p><code class='latex inline'>\displaystyle{\log_2\frac{1}{4}-\log_31}</code></p>
<p>Express in exponential form.</p><p><code class='latex inline'>\displaystyle{\log_5\frac{1}{25}=-2}</code></p>
<p> Evaluate the following using the property <code class='latex inline'>\displaystyle \log_a a^x = x</code></p><p><code class='latex inline'> \displaystyle \log_9 27^2 </code></p>
<p>Find the value of <code class='latex inline'>-\frac{1}{3}\log_{2}8</code>.</p>
<p>Evaluate.</p><p><code class='latex inline'>\log_6\sqrt{6}</code></p>
<p>Evaluate.</p><p><code class='latex inline'>\log_381+\log_464</code></p>
<p>Why can <code class='latex inline'>\log_3(-9)</code> not be evaluated?</p>
<p>Simplify each expression.</p><p><code class='latex inline'>\displaystyle \ln e </code></p>
<p>Determine the inverse of each relation.</p><p><code class='latex inline'>y=3\log_2(x-3)+2</code></p>
<p> Evaluate the following using the property <code class='latex inline'>\displaystyle \log_a a^x = x</code></p><p><code class='latex inline'> \displaystyle \log_981 </code></p>
<p>Express in exponential form.</p><p><code class='latex inline'>\displaystyle{\log_{\frac{1}{6}}216=-3}</code></p>
<p> Change the following to logarithmic notation, <code class='latex inline'>x = a^y</code>.</p><p><code class='latex inline'> \displaystyle \begin{array}{ccccc} &a) & \log_{4}1024 = 5 &b) & \log_{7}7 = 1 \\ &c) & \log_{6}7776 = 5 &d) & \log_{a}p = r \end{array} </code></p>
<p> Evaluate the expression.</p><p><code class='latex inline'> \displaystyle \log_{10}(100) </code></p>
<p>The doubling function <code class='latex inline'>\displaystyle{y=y_02^{\frac{t}{D}}}</code> can be used to model exponential growth when the doubling time is <code class='latex inline'>D</code>. The bacterium <code class='latex inline'>Escherichia</code> <code class='latex inline'>coli</code> has a doubling period of 0.32 h. A culture of <code class='latex inline'>E.</code> <code class='latex inline'>coli</code> starts with 100 bacteria.</p> <ul> <li>How many hours will it take for there to be 450 bacteria in the culture? Explain your strategy.</li> </ul>
<p>Isolate x.</p><p><code class='latex inline'> \begin{array}{ccccccc} &(a) & 3^x = 32 & (b)& 5^x = 66 \\ &(c)&10^x = 157 &(d) &1.03^x = 1.69 \\ &(e) & 4.5^x = 600 &(f) & \log 1.07^x = 2 \\ \end{array} </code></p>
<p>The doubling function <code class='latex inline'>\displaystyle{y=y_02^{\frac{t}{D}}}</code> can be used to model exponential growth when the doubling time is <code class='latex inline'>D</code>. The bacterium <code class='latex inline'>Escherichia</code> <code class='latex inline'>coli</code> has a doubling period of 0.32 h. A culture of <code class='latex inline'>E</code>. <code class='latex inline'>coli</code> starts with 100 bacteria.</p> <ul> <li>Determine the equation for the number of bacteria, <code class='latex inline'>y</code>, in <code class='latex inline'>x</code> hours.</li> </ul>
<p> Evaluate the expression.</p><p><code class='latex inline'> \displaystyle \log_7(\log 10^{49}) </code></p>
<p>The doubling function <code class='latex inline'>\displaystyle{y=y_02^{\frac{t}{D}}}</code> can be used to model exponential growth when the doubling time is <code class='latex inline'>D</code>. The bacterium <code class='latex inline'>Escherichia</code> <code class='latex inline'>coli</code> has a doubling period of 0.32 h. A culture of <code class='latex inline'>E.</code> <code class='latex inline'>coli</code> starts with 100 bacteria.</p> <ul> <li>Determine the equation of the inverse. What does this equation represent?</li> </ul>
<p>Determine the inverse of each relation.</p> <ul> <li><code class='latex inline'>y=\sqrt[3]{x}</code></li> </ul>
<p> Evaluate the following using the property <code class='latex inline'>a^{\log_a x} = x</code></p><p><code class='latex inline'> \displaystyle 2^{2\log_23} </code></p>
<p>Graph the function and its inverse. State the domain, range, and asymptote of each. Determine the equation of the inverse.</p><p><code class='latex inline'>y=2+3\log x</code></p>
<p>Evaluate without a calculator.</p><p><code class='latex inline'> \displaystyle 81^{0.5 \log_9 7} </code></p>
<p>Solve for x.</p><p><code class='latex inline'> \displaystyle \log_{0.04}5 = x </code></p>
<p> Evaluate the expression.</p><p><code class='latex inline'> \displaystyle \log_{3}\sqrt{27} </code></p>
<p>Evaluate.</p><p> <code class='latex inline'>\log_3\sqrt{27}</code></p>
<p>Evaluate <code class='latex inline'>\log_216^{\frac{1}{3}}</code>.</p>
<p>Solve for <code class='latex inline'>x</code>. Round your answers to two decimal places, if necessary.</p><p><code class='latex inline'>\log x=-2</code></p>
<p>Evaluate without a calculator.</p><p><code class='latex inline'> \displaystyle 5^{\log_5 10 - 1} </code></p>
<p>Graph the function and its inverse. State the domain, range, and asymptote of each. Determine the equation of the inverse.</p><p><code class='latex inline'>y=3\log(x+6)</code></p>
<p> Evaluate the following without a calculator</p><p><code class='latex inline'> \displaystyle \log_3 27^2 </code></p>
<p>Evaluate without a calculator.</p><p><code class='latex inline'> \displaystyle 36^{\log_6 3} </code></p>
<p>Simplify each expression.</p><p><code class='latex inline'>\displaystyle \frac{\ln e^{2}}{2} </code></p>
<p>The number of mold spores in a petri dish increases by a factor of 10 every week. If there are initially 40 spores in the dish, how long will it take for there to be 2000 spores?</p>
<p> Evaluate the expression.</p><p><code class='latex inline'> \displaystyle \begin{array}{cccc} &(i)&\log_{7}1 &(ii) & \log_{4}{1024} \\ \end{array} </code></p>
<p>Determine the inverse of each relation.</p><p><code class='latex inline'>y=3(2)^x</code></p>
<p>Consider the expression <code class='latex inline'>\log_5a</code>.</p> <ul> <li>For what values of <code class='latex inline'>a</code> will this expression be undefined?</li> </ul>
<p>Solve for <code class='latex inline'>x</code>. Round your answers to two decimal places, if necessary.</p><p><code class='latex inline'>\log25=x</code></p>
<p>Solve for <code class='latex inline'>x</code>. Round your answers to two decimal places, if necessary.</p><p><code class='latex inline'>\log x=0.25</code></p>
<p>Simplify each expression.</p><p><code class='latex inline'>\displaystyle \ln 1 </code></p>
<p> Evaluate the following using the property <code class='latex inline'>\displaystyle \log_a a^x = x</code></p><p><code class='latex inline'> \displaystyle \log_{121}11 </code></p>
<p> Change the following to logarithmic notation, <code class='latex inline'>y = \log_{a}x</code>.</p><p><code class='latex inline'> \begin{array}{ccccc} &(a) & 3^5 = 243 &(b) & 2^7 = 128 \\ &(c) & 7^3 = 343 &(d) & 11^4 = 14 641 \end{array} </code></p>
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