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Similar Question 1
<p>Express in exponential form.</p><p>c) <code class='latex inline'>\log_381=4</code></p>
Similar Question 2
<p>Write <code class='latex inline'>\frac{1}{2} \log_ax + \frac{1}{2}\log_ay - \frac{3}{4} \log_az</code> as a single logarithm. Assume that all the variables represent positive numbers.</p>
Similar Question 3
<p>What is the value of <code class='latex inline'>\displaystyle \log _{5}\left(\frac{1}{125}\right) ? </code> </p><p><code class='latex inline'>\displaystyle \begin{array}{llll}\text { A }-3 & \text { B }-2 & \text { C } 2 & \text { D } 25\end{array} </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>Express each as a single logarithm. Then, evaluate, if possible. </p><p><code class='latex inline'>\displaystyle \log_4192 -\log_4 3 </code></p>
<p>Simplify each algebraic expression. State any restrictions on the variables.</p><p><code class='latex inline'>\log x + \log y + \log (2z)</code></p>
<p>Express in logarithmic form.</p><p>e) <code class='latex inline'>\displaystyle{\left(\frac{1}{3}\right)^3=\frac{1}{27}}</code></p>
<p>Estimate the value of each of the following logarithms to two decimal places.</p><p>a) <code class='latex inline'>\log_432</code></p>
<p>Evaluate.</p><p><code class='latex inline'>\displaystyle \log_7343 </code></p>
<p>Express as a single logarithm.</p><p><code class='latex inline'>\displaystyle \log 5 - \log 2 </code></p>
<p>Express <code class='latex inline'>\log_4x^2 + 3\log_4y\frac{1}{3} - \log_4x</code> as a single logarithm. Assume that <code class='latex inline'>x</code> and <code class='latex inline'>y</code> represent positive numbers.</p>
<p>Write each expression as a single log. Assume that all the variables represent positive numbers</p><p><code class='latex inline'> \displaystyle \log_6a - (\log_6 b + \log_6 c) </code></p>
<p>Express in exponential form.</p><p>e) <code class='latex inline'>\displaystyle{\log_6\sqrt{6}=\frac{1}{2}}</code></p>
<p>Evaluate</p><p><code class='latex inline'>\displaystyle \log_3 9\sqrt{27} </code></p>
<p>Graph the function <code class='latex inline'>q(x) = \log x^5</code>.</p>
<p>Evaluate, using the product law of logarithms.</p><p><code class='latex inline'>\displaystyle \log 5 + \log 40 + \log 9 </code></p>
<p>At a concert, the loudness of sound, L, in decibels, is given by the equation <code class='latex inline'>L = 10 \log\frac{I }{I_0}</code>, where <code class='latex inline'>I</code> is the intensity, in watts per square metre, <code class='latex inline'>I_0</code> and the minimum intensity of sound audible to the average person, or <code class='latex inline'>1.0 \times 10^{-12} W/m^2</code>.</p> <ul> <li>On the way home from the concert,your car stereo produces 120 dB of sound. What is its intensity?</li> </ul>
<p>Evaluate.</p><p>c) <code class='latex inline'>\displaystyle{\log_2\left(\frac{1}{4}\right)}</code></p>
<p>Write as a sum or difference of logarithms. Simplify, if possible.</p><p><code class='latex inline'>\displaystyle \log_{3}(\frac{m}{n}) </code></p>
<p>Express in logarithmic form.</p><p>f) <code class='latex inline'>\displaystyle{8^{\frac{1}{3}}=2}</code></p>
<p>Evaluate, correct to three decimal places.</p><p><code class='latex inline'>\displaystyle \log _{6} 27 </code></p>
<p>Evaluate using log laws.</p><p><code class='latex inline'>\displaystyle \log_642 - \log_67 </code></p>
<p>Express in exponential form.</p><p>a) <code class='latex inline'>\log_28=3</code></p>
<p>Evaluate </p><p><code class='latex inline'>\displaystyle \log_{\frac{2}{3}} \frac{27}{8} </code></p>
<p>How are the functions <code class='latex inline'>p(x) = 2 \log x + 3 \log x</code> and <code class='latex inline'>q(x) = \log x^5</code> related? What law of logarithms does this illustrate?</p>
<p>Write as a sum or difference of logarithms. Simplify, if possible.</p><p><code class='latex inline'>\displaystyle \log(uv^3) </code></p>
<p>Write as a single logarithm.</p><p><code class='latex inline'> \displaystyle 2\log a + \log(3b) - \frac{1}{2}\log c </code></p>
<p>Determine the inverse of the function. Express your answers in log form.</p><p><code class='latex inline'>\displaystyle m = p^q </code></p>
<p>Write as a sum or difference of logarithms. Simplify, if possible.</p><p><code class='latex inline'> \displaystyle \log(a^2bc) </code></p>
<p>Determine the inverse of the function. Express your answers in log form.</p><p><code class='latex inline'>\displaystyle y = 4^x </code></p>
<p>Evaluate.</p><p><code class='latex inline'>\displaystyle \log_{\frac{1}{5}}25 </code></p>
<p>Write as a sum or difference of logarithms. Simplify, if possible.</p><p><code class='latex inline'>\displaystyle \log(\frac{a\sqrt{b}}{c^2}) </code></p>
<p>Simplify each algebraic expression. State any restrictions on the variables.</p><p><code class='latex inline'>2 \log u + \log v + \frac{1}{2}\log w</code></p>
<p>Evaluate, using the product law of logarithms.</p><p><code class='latex inline'>\displaystyle \log_{4} 320 + \log_{4}5 </code></p>
<p>Express each of the following in the form <code class='latex inline'>r \log_ax</code>.</p><p><code class='latex inline'> \displaystyle \log_5\sqrt{5}[125] </code></p>
<p>Write each expression as a single log. Assume that all the variables represent positive numbers</p><p><code class='latex inline'> \displaystyle 1 +\log_3x^2 </code></p>
<p>Write each expression with base 3.</p><p><code class='latex inline'>\displaystyle \frac{1}{2} </code></p>
<p>Evaluate</p><p><code class='latex inline'>\displaystyle \log_714 + \log_73.5 </code></p>
<p>Evaluate.</p><p><code class='latex inline'>\displaystyle \log_{19}1 </code></p>
<p>Write as a sum or difference of logarithms. Simplify, if possible.</p><p><code class='latex inline'>\displaystyle \log_550 </code></p>
<p>Express in exponential form.</p><p>c) <code class='latex inline'>\log_381=4</code></p>
<p>Evaluate </p><p><code class='latex inline'>\displaystyle \log_51 </code></p>
<p>Write <code class='latex inline'>\frac{1}{2} \log_ax + \frac{1}{2}\log_ay - \frac{3}{4} \log_az</code> as a single logarithm. Assume that all the variables represent positive numbers.</p>
<p>Write it as a power of 5.</p> <ul> <li>20</li> </ul>
<p>Express in logarithmic form.</p><p>d) <code class='latex inline'>\displaystyle{6^{-2}=\frac{1}{36}}</code></p>
<p>Write it as a power of 5.</p> <ul> <li>0.8</li> </ul>
<p>Evaluate, using the product law of logarithms.</p><p><code class='latex inline'>\displaystyle \log_354 - \log_32 </code></p>
<p>Graph the sum of these two functions: <code class='latex inline'>p(x) = f(x) + g(x)</code> where <code class='latex inline'>f(x) = 2 \log x</code> and <code class='latex inline'>g(x) = 3 \log x</code>.</p>
<p>Evaluate</p><p><code class='latex inline'>\displaystyle \log_4\sqrt[3]{16} </code></p>
<p>Write each expression as a single log. Assume that all the variables represent positive numbers</p><p><code class='latex inline'> \displaystyle \log_2x^2 - \log_2xy + \log_2y^2 </code></p>
<p>Express as a single logarithm.</p><p><code class='latex inline'>\displaystyle \log_56 + \log_58-\log_512 </code></p>
<p>What is the value of <code class='latex inline'>\displaystyle \log _{4} 16^{-3} </code> ? </p><p><code class='latex inline'>\displaystyle \begin{array}{llll}\text { A } \frac{1}{8} & \text { B }-6 & \text { C }-12 & \text { D undefined }\end{array} </code></p>
<p>Write as a sum or difference of logarithms. Simplify, if possible.</p><p><code class='latex inline'>\displaystyle \log_7(cd) </code></p>
<p>Draw the graphs of <code class='latex inline'>y = \log x + \log 2x</code> and <code class='latex inline'>y =\log 2x^2</code>. Although the graph are different, simplifying the first expression using the laws of logarithms produces the second expression. Explain why the graph are different.</p>
<p>Evaluate, using the product law of logarithms.</p><p> <code class='latex inline'>\displaystyle 3\log_{16}2 + 2\log_{16}8 - \log_{16}2 </code></p>
<p>Evaluate the value of each expression to three decimal places.</p><p><code class='latex inline'>\displaystyle \begin{array}{llllllll} &(a) \phantom{.} \log_221 &(b) \phantom{.} \log_5117 \\ &(c) \phantom{.} \log_7141 &(d) \phantom{.} \log_{11}356 \end{array} </code></p>
<p>Evaluate.</p><p>b) <code class='latex inline'>\log_71</code></p>
<p>Simplify and state any restrictions on the variables.</p><p><code class='latex inline'> \displaystyle \log(2m + 6) -\log(m^2 - 9) </code></p>
<p>Evaluate </p><p><code class='latex inline'>\displaystyle \log_381 </code></p>
<p>Simplify. State any restrictions on the variables.</p><p><code class='latex inline'>\displaystyle \log(x^2 + 7x + 12 ) -\log(x^2 - 9) </code></p>
<p>Simplify, using the laws of logarithms.</p><p><code class='latex inline'>\log_3 7 + \log_3 3</code></p>
<p>Evaluate.</p><p>e) <code class='latex inline'>\displaystyle{\log_{\frac{2}{3}}\left(\frac{8}{27}\right)}</code></p>
<p>Evaluate, using the product law of logarithms.</p><p> <code class='latex inline'>\displaystyle \log 20 + \log 2 +\frac{1}{3}\log 125 </code></p>
<p>Is the following statement true?</p><p><code class='latex inline'> \displaystyle \log(-3) + \log(-4) =\log 12 </code></p>
<p>Express as a single logarithm.</p><p><code class='latex inline'>\displaystyle \log_311 + \log_34 -\log _36 </code></p>
<p>Evaluate, using the product law of logarithms.</p><p><code class='latex inline'>\displaystyle \log 50 000 + \log 5 </code></p>
<p>Evaluate</p><p><code class='latex inline'>\displaystyle \log_{\frac{1}{2}}72 - \log_{\frac{1}{2}}9 </code></p>
<p>What is the value of <code class='latex inline'>\displaystyle \log _{5}\left(\frac{1}{125}\right) ? </code> </p><p><code class='latex inline'>\displaystyle \begin{array}{llll}\text { A }-3 & \text { B }-2 & \text { C } 2 & \text { D } 25\end{array} </code></p>
<p>Express as a single logarithm.</p><p><code class='latex inline'>\displaystyle \log 7+ \log 4 </code></p>
<p>A certain operational amplifier (Op Amp) produces a voltage output, <code class='latex inline'>V_0</code>, in volts, from two input voltage signals, <code class='latex inline'>V_1</code> and <code class='latex inline'>V_2</code>, according to the equation <code class='latex inline'>V_0 = \log V_2 ? \log V_1</code>.</p> <ul> <li>Write a simplified form of this formula, expressing the right side as a single logarithm.</li> </ul>
<p>Write as a sum or difference of logarithms. Simplify, if possible.</p><p><code class='latex inline'> \displaystyle \log(\frac{k}{\sqrt{m}}) </code></p>
<p>Write each expression as a single power of 4.</p><p><code class='latex inline'>(\sqrt{14})^3</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 travels a distance of about 50 miles, estimate its wind speed near its centre.</li> </ul>
<p>Explain how the laws of logarithms can help you evaluate <code class='latex inline'>\log_3 (\frac{\sqrt[5]{27}}{2187})</code>.</p>
<p>Write the expression as a sum of difference of logarithms.</p><p><code class='latex inline'> \displaystyle \log_mpq </code></p>
<p>Evaluate using log laws.</p><p><code class='latex inline'>\displaystyle \log_7 \sqrt[3]{49} </code></p>
<p>Express as a single logarithm.</p><p><code class='latex inline'>\displaystyle 2\log 3 + 4\log 2 </code></p>
<p>Evaluate, using the product law of logarithms.</p><p><code class='latex inline'>\displaystyle \log 2 + \log 200 </code></p>
<p>Simplify each algebraic expression. State any restrictions on the variables.</p><p><code class='latex inline'>2\log m + 3\log n - 4\log y</code></p>
<p>Express in logarithmic form.</p><p> <code class='latex inline'>3^4=81</code></p>
<p>Write each expression with base 2.</p><p><code class='latex inline'>\displaystyle 14 </code></p>
<p>Express each as a single logarithm. Then, evaluate, if possible. </p><p><code class='latex inline'>\displaystyle \log_a(ab) -\log_a(a^3b) </code></p>
<p>Evaluate without a calculator.</p><p><code class='latex inline'> \displaystyle \log_4128 - \log_48 </code></p>
<p>Express each as a single logarithm. Then, evaluate, if possible. </p><p><code class='latex inline'>\displaystyle \log_535 -\log_5 7 + \log_525 </code></p>
<p>A certain operational amplifier (Op Amp) produces a voltage output, <code class='latex inline'>V_0</code>, in volts, from two input voltage signals, <code class='latex inline'>V_1</code> and <code class='latex inline'>V_2</code>, according to the equation <code class='latex inline'>V_0 = \log V_2 ? \log V_1</code>.</p><p> What is the voltage output if</p> <ul> <li>i) <code class='latex inline'>V_2</code> is 10 times <code class='latex inline'>V_1</code>?</li> <li>ii) <code class='latex inline'>V_2</code> is 100 times <code class='latex inline'>V_1</code>? </li> <li>ii) <code class='latex inline'>V_2</code> is equal to <code class='latex inline'>V_1</code>? </li> </ul>
<p>Estimate the value to three decimal places.</p><p><code class='latex inline'>\displaystyle \begin{array}{llllllll} &(a) \phantom{.} \log_353 &(b) \phantom{.} \log_4 \frac{1}{10} \\ &(c) \phantom{.} \log_6 159 &(d) \phantom{.} \log_{15} 1456 \end{array} </code></p>
<p>Express each as a single logarithm. Then, evaluate, if possible. </p><p><code class='latex inline'>\displaystyle \log(xy) + \log\frac{y}{x} </code></p>
<p>Evaluate, using the product law of logarithms.</p><p><code class='latex inline'>\displaystyle \log_{12} 8 + \log_{12}2 + \log_{12}9 </code></p>
<p>Evaluate the three decimal places.</p><p><code class='latex inline'>\displaystyle \begin{array}{llllllll} &(a) \phantom{.} \log 4 &(b) \phantom{.} \log 45 \\ &(c) \phantom{.} \log 135 &(d) \phantom{.} \log 300 \end{array} </code></p>
<p>Simplify, using the laws of logarithms.</p><p><code class='latex inline'>\log 48 - \log 6</code></p>
<p>Determine the inverse of the function. Express your answers in log form.</p><p><code class='latex inline'>\displaystyle y = (\frac{3}{4})^x </code></p>
<p>Graph the functions <code class='latex inline'>f(x) = 2 \log x</code> and <code class='latex inline'>g(x) = 3 \log x</code>.</p>
<p>Simplify, using the laws of logarithms.</p><p><code class='latex inline'>\log 9 + \log 6</code></p>
<p>Evaluate.</p><p>d) <code class='latex inline'>\log_7\sqrt{7}</code></p>
<p>Write as a sum or difference of logarithms. Simplify, if possible.</p><p><code class='latex inline'>\displaystyle \log_210 </code></p>
<p>Simplify. State any restrictions on the variables.</p><p><code class='latex inline'>\displaystyle 2\log w + 3\log\sqrt{w} + \frac{1}{2}\log w^2 </code></p>
<p>Write each expression as a single log. Assume that all the variables represent positive numbers</p><p><code class='latex inline'> \displaystyle 3 \log_4 x + 2 \log_4x - \log_4 y </code></p>
<p>For the function <code class='latex inline'> y=\log x </code>, where <code class='latex inline'>0 < x < 1000</code>, how many integer values of <code class='latex inline'>y</code> are possible if <code class='latex inline'>y>-20</code>?</p>
<p>Write as a single logarithm.</p><p><code class='latex inline'> \displaystyle \log_78 + \log_74 - \log_716 </code></p>
<p>Simplify. State any restrictions on the variables.</p><p><code class='latex inline'>\displaystyle \log(\frac{x^2}{\sqrt{x}}) </code></p>
<p>Write each expression with base 3.</p><p>10</p>
<p>Express as a single logarithm.</p><p><code class='latex inline'>\displaystyle \log 5 + \log 11 </code></p>
<p>Determine the inverse of the function. Express your answers in log form.</p><p><code class='latex inline'>\displaystyle y = a^x </code></p>
<p>Simplify each algebraic expression. State any restrictions on the variables.</p><p><code class='latex inline'>\log_2a + \log_2(3b) - \log_2(2c)</code></p>
<p>Simplify, using the laws of logarithms.</p><img src="/qimages/14253" />
<p>Simplify. State any restrictions on the variables.</p><p> <code class='latex inline'>\displaystyle \log(\frac{\sqrt{m}}{m^3}) +\log(\sqrt{m})^7 </code></p>
<p>Evaluate.</p><p>a) <code class='latex inline'>\displaystyle{\log_33^5}</code></p>
<p>Is the following statement true? <code class='latex inline'> \displaystyle -\log 3 - \log 4 = -\log 12 </code></p><p>Explain why or why not.</p>
<p>Simplify. State any restrictions on the variables.</p><p> <code class='latex inline'>\displaystyle \log(x^2 -4) - \log(x - 2) </code></p>
<p>Write each expression as a single log. Assume that all the variables represent positive numbers</p><p><code class='latex inline'> \displaystyle \log_5 u - \log_5 v +\log_5 w </code></p>
<p>Evaluate.</p><p>f) <code class='latex inline'>\log_2\sqrt[3]{2}</code></p>
<p>Express as a single logarithm.</p><p><code class='latex inline'>\displaystyle \log_pq + \log_pq </code></p>
<p> Use the power law of logarithms to verify the product law of logarithms for <code class='latex inline'>\log 10^2</code>.</p>
<p>Evaluate, using the product law of logarithms.</p><p><code class='latex inline'>\displaystyle \log_618 + \log_62 </code></p>
<p>Express as a single logarithm.</p><p><code class='latex inline'>\displaystyle \log 20 - \log 4 </code></p>
<p>Explain why <code class='latex inline'>\log_xx^{m - 1} + 1 = m</code>.</p>
<p>Evaluate, using the product law of logarithms.</p><p><code class='latex inline'>\displaystyle \log 40 + \log 2.5 </code></p>
<p>Show that if </p> <ul> <li><code class='latex inline'>\log_ba = c</code> and</li> <li><code class='latex inline'> \log_yb = c</code> then</li> <li><code class='latex inline'>\log_4y =</code>?</li> </ul>
<p>Use the laws of logarithms to express each of the following int terms of <code class='latex inline'>\log_b x , \log_by</code>, and <code class='latex inline'>\log_bz</code>.</p><p><code class='latex inline'> \displaystyle \log_b(\frac{z}{xy}) </code></p>
<p>Simplify. State any restrictions on the variables.</p><p><code class='latex inline'>\displaystyle \log \sqrt{k} + \log(\sqrt{k})^3 + \log \sqrt[3]{k^2} </code></p>
<p><strong>a)</strong> Explain how you can transform the graph of <code class='latex inline'>f(x) = \log x</code> to produce <code class='latex inline'>g(x) = \log (10nx)</code>, for any <code class='latex inline'>n > 0</code>.</p><p><strong>b)</strong> Create two examples to support your explanation. Sketch graphs to illustrate.</p>
<p>Simplify. State any restrictions on the variables.</p><p><code class='latex inline'>\displaystyle \log(x^2 - x -6) - \log(2x - 6) </code></p>
<p>Evaluate using log laws.</p><p><code class='latex inline'>\displaystyle 2 \log_48 </code></p>
<p>Given the formula from Example 1 for the magnitude of an earthquake, <code class='latex inline'>\displaystyle{R=\log\left(\frac{a}{T}\right)+B}</code>, determine the value of <code class='latex inline'>a</code> if <code class='latex inline'>R=6.3</code>, and <code class='latex inline'>T=1.6</code></p>
<p>Evaluate.</p><p><code class='latex inline'>\displaystyle \begin{array}{llllllll} &(a) \phantom{.} \log_3 \frac{1}{9} &(b) \phantom{.} \log_{5}100 - \log_54 \end{array} </code></p>
<p>Express in exponential form.</p><p>f) <code class='latex inline'>\log_{10}1=0</code></p>
<p>Evaluate using log laws.</p><p><code class='latex inline'>\displaystyle \log_35 + \log_318 -\log_310 </code></p>
<p>Evaluate.</p><p><code class='latex inline'>\displaystyle \begin{array}{llllllll} &(a) \phantom{.} \log 15 + \log 40 - \log 6 \\ &(b) \phantom{.} \log_7343 = 2\log_749 \end{array} </code></p>
<p>Use the laws of logarithms to express each of the following int terms of <code class='latex inline'>\log_b x , \log_by</code>, and <code class='latex inline'>\log_bz</code>.</p><p><code class='latex inline'> \displaystyle \log_b\sqrt{x^5yz^3} </code></p>
<p>Evaluate.</p><p><code class='latex inline'>\displaystyle \log_{4}\frac{1}{256} </code></p>
<p>Simplify and state any restrictions on the variables.</p><p><code class='latex inline'> \displaystyle \log(x^2 + 2x - 15) -\log(x^2 - 7x + 12) </code></p>
<p>Express in logarithmic form.</p><p> <code class='latex inline'>4^2=16</code></p>
<p>Evaluate.</p><p>a) <code class='latex inline'>\log_55</code></p>
<p>Evaluate</p><p><code class='latex inline'>\displaystyle \log_5100 + \log_5 \frac{1}{4} </code></p>
<p>Evaluate </p><p><code class='latex inline'>\displaystyle \log_4 \frac{1}{16} </code></p>
<p>Explain why <code class='latex inline'>\log_53 + \log_5\frac{1}{3} = 0</code>.</p>
<p>Write each expression as a single log. Assume that all the variables represent positive numbers</p><p><code class='latex inline'> \displaystyle \log_2 x +\log_2 y + \log_2 z </code></p>
<p>Evaluate without a calculator.</p><p><code class='latex inline'> \displaystyle \log_68 + \log_627 </code></p>
<p>Use the laws of logarithms to express each of the following int terms of <code class='latex inline'>\log_b x , \log_by</code>, and <code class='latex inline'>\log_bz</code>.</p><p><code class='latex inline'> \displaystyle \log_bx^2y^3 </code></p>
<p>Given the parent function <code class='latex inline'>y = \log x</code>, write the equation of the function that results from each set of transformations.</p><p>a) vertical stretch by a factor of 4, followed by a reflection in the x-axis</p><p>b) horizontal translation 3 units to the left followed by a vertical translation 1 unit up.</p><p>c) vertical compression by a factor of <code class='latex inline'>\frac{2}{3}</code>, followed by a horizontal stretch by a factor of <code class='latex inline'>2</code>.</p><p>d) vertical stretch a factor of 3, followed y a reflection in the y-axis and a horizontal translation 1 unit to the right.</p>
<p>Simplify. State any restrictions on the variables.</p><p> <code class='latex inline'>\displaystyle \log(x^2 + 7x +12) - \log(x +3) </code></p>
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