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Wize High School Grade 12 Pre-Calculus Textbook > Logarithmic Functions

Logarithm Laws

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Logarithm Laws

Law of Logarithms for Real Numbers

If mm and nn are any real number, then:

1.log⁡bx+log⁡by=log⁡b(xy)2.log⁡bx−log⁡by=log⁡b(xy)3.log⁡b(x)n=n⋅log⁡bx4.log⁡bb=15.log⁡bbn=n6.log⁡b1=07.log⁡ba=log⁡calog⁡cbBase Change Formula\begin{array}{lccl} 1.&\log_{b}{x}+\log_{b}{y}&=&\log_{b}{(xy)}\\\\ 2.&\log_{b}{x}-\log_{b}{y}&=&\log_{b}{\Big(\dfrac{x}{y}\Big)}\\\\ 3.&\log_{b}{(x)}^n&=&n\cdot\log_{b}{x}\\\\ 4.&\log_{b}{b}&=&1\\\\ 5.&\log_{b}{b}^n&=&n\\\\ 6.&\log_{b}{1}&=&0\\\\ 7.&\log_{b}{a}&=&\dfrac{\log_{c}{a}}{\log_{c}{b}}&\colorTwo{\footnotesize{\text{Base Change Formula}}} \end{array}

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Example

Write as a single logarithm: 6log⁡2x+12log⁡2y−3log⁡2z6\log_{2}{x}+\dfrac{1}{2}\log_{2}{y}-3\log_{2}{z}

6log⁡2x+12log⁡2y−3log⁡2z=log⁡2x6+log⁡2y1/2−log⁡2z3=log⁡2x6+log⁡2y−log⁡2z3=log⁡2x6yz3\begin{array}{rcl} 6\log_{2}{x}+\dfrac{1}{2}\log_{2}{y}-3\log_{2}{z}&=&\log_{2}{x^6}+\log_{2}{y^{1/2}}-\log_{2}{z^3}\\\\ &=&\log_{2}{x^6}+\log_{2}{\sqrt{y}}-\log_{2}{z^3}\\\\ &=&\log_{2}{\dfrac{x^6\sqrt{y}}{z^3}} \end{array}
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Example: Logarithm Laws

Simplify 2log⁡b3a−23log⁡bb+35log⁡b4c2\log_{b}{3a}-\dfrac{2}{3}\log_{b}{b}+\dfrac{3}{5}\log_{b}{4c}.


2log⁡b3a−23log⁡bb+35log⁡b4c=log⁡b(3a)2−log⁡bb2/3+log⁡b(4c)3/5=log⁡b(9a2(4c)3/5b2/3)\begin{array}{rcl} 2\log_{b}{3a}-\dfrac{2}{3}\log_{b}{b}+\dfrac{3}{5}\log_{b}{4c}&=&\log_{b}{(3a)^2}-\log_{b}{b^{2/3}}+\log_{b}{(4c)^{3/5}}\\\\ &=&\log_{b}{\Bigg(\dfrac{9a^2(4c)^{3/5}}{b^{2/3}}\Bigg)} \end{array}
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Example: Logarithm Laws

Expand log⁡(24x2y317z4)\log_{}{\Bigg(\dfrac{24x^2y^3}{17z^4}\Bigg)}.


log⁡(24x2y317z4)=log⁡24x2y3−log⁡17z4=log⁡24+log⁡x2+log⁡y3−(log⁡17+log⁡z4)\begin{array}{rcl} \log_{}{\Bigg(\dfrac{24x^2y^3}{17z^4}\Bigg)}&=&\log_{}{24x^2y^3}-\log_{}{17z^4}\\\\ &=&\log_{}{24+\log_{}{x^2}}+\log_{}{y^3}-(\log_{}{17}+\log_{}{z^4}) \end{array}

Practice: Logarithm Laws

Which of the following is equivalent to log⁡2(16mn6p4s3)\log_{2}{\Bigg(\dfrac{16\sqrt{m}n^6}{p^4s^3}\Bigg)}?

Practice: Logarithm Laws

If log⁡2=a\log2=a and log⁡3=b\log3=b, write each logarithm in terms of aa and bb.


Practice: Logarithm Laws

If log⁡70=1.8451\log70=1.8451, find an approximation for the following:
Extra Practice
Logarithmic Functions
If log⁡3=a\log{3}=a and log⁡5=b\log{5}=b, then determine the following in terms of a and b:
Logarithm Laws
Practice: Logarithm Laws
If log⁡50=9.5\log50=9.5, find an approximation for the following:
Logarithm Laws
Practice: Logarithm Laws
If log⁡20=3.4872\log20=3.4872, find an approximation for the following:
Logarithm Laws
Practice: Logarithm Laws
If log⁡5=a\log5=a and log⁡7=b\log7=b, write each logarithm in terms of aa and bb.
Logarithm Laws
Practice: Logarithm Laws
If log⁡3=a\log3=a and log⁡5=b\log5=b, write each logarithm in terms of aa and bb.
Logarithm Laws
Practice: Logarithm Laws
Evaluate log⁡3(812734353965)\log_{3}{\Bigg(\dfrac{81\sqrt[4]{27^3}3^{\frac{5}{3}}}{9^{\frac{6}{5}}}\Bigg)}. Leave answer in exact form.
Logarithm Laws
Practice: Logarithm Laws
Which of the following is equivalent to log⁡a(a4b3c5d32e2)\log_{a}{\Bigg(\dfrac{a^4\sqrt[3]{b}c^5}{d^{\frac{3}{2}}e^2}\Bigg)}?