Wize High School Algebra II Textbook (Common Core) > Logarithmic Functions

Logarithm Laws

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Logarithm Laws
Law of Logarithms for Real Numbers
If mmm and nnn 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}1.2.3.4.5.6.7.​logb​x+logb​ylogb​x−logb​ylogb​(x)nlogb​blogb​bnlogb​1logb​a​=======​logb​(xy)logb​(yx​)n⋅logb​x1n0logc​blogc​a​​Base Change Formula​

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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}6log2​x+21​log2​y−3log2​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}6log2​x+21​log2​y−3log2​z​===​log2​x6+log2​y1/2−log2​z3log2​x6+log2​y​−log2​z3log2​z3x6y​​​

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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}2logb​3a−32​logb​b+53​logb​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}2logb​3a−32​logb​b+53​logb​4c​==​logb​(3a)2−logb​b2/3+logb​(4c)3/5logb​(b2/39a2(4c)3/5​)​

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Example: Logarithm Laws
Expand log⁡(24x2y317z4)\log_{}{\Bigg(\dfrac{24x^2y^3}{17z^4}\Bigg)}log​(17z424x2y3​).

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}log​(17z424x2y3​)​==​log​24x2y3−log​17z4log​24+log​x2+log​y3−(log​17+log​z4)​

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)}log2​(p4s316m​n6​)?

A) 4log⁡22+12log⁡2m+6log⁡2n−4log⁡2p−3log⁡2s4\log_{2}{2}+\dfrac{1}{2}\log_{2}{m}+6\log_{2}{n}-4\log_{2}{p}-3\log_{2}{s}4log2​2+21​log2​m+6log2​n−4log2​p−3log2​s 

B) 4log⁡22+12log⁡2m+6log⁡2n−4log⁡2p+3log⁡2s4\log_{2}{2}+\dfrac{1}{2}\log_{2}{m}+6\log_{2}{n}-4\log_{2}{p}+3\log_{2}{s}4log2​2+21​log2​m+6log2​n−4log2​p+3log2​s 

C) 4log⁡22+12log⁡2m+6log⁡2n+4log⁡2p+3log⁡2s4\log_{2}{2}+\dfrac{1}{2}\log_{2}{m}+6\log_{2}{n}+4\log_{2}{p}+3\log_{2}{s}4log2​2+21​log2​m+6log2​n+4log2​p+3log2​s 

D) 4log⁡22+12log⁡2m+6log⁡2n+4log⁡2p−3log⁡2s4\log_{2}{2}+\dfrac{1}{2}\log_{2}{m}+6\log_{2}{n}+4\log_{2}{p}-3\log_{2}{s}4log2​2+21​log2​m+6log2​n+4log2​p−3log2​s  

I don't know

Practice: Logarithm Laws
If log⁡2=a\log2=alog2=a and log⁡3=b\log3=blog3=b, write each logarithm in terms of aaa and bbb.

\log6

\log1.5

\log60

\log12

\log18

\log36

\log3.6

\log{\Bigg(\dfrac{1}{6}\Bigg)}

I don't know

Practice: Logarithm Laws
If log⁡70=1.8451\log70=1.8451log70=1.8451, find an approximation for the following:

\log7

\log0.7

I don't know

Extra Practice

Logarithmic Functions

If log⁡3=a\log{3}=alog3=a and log⁡5=b\log{5}=blog5=b, then determine the following in terms of a and b:

Logarithm Laws

Practice: Logarithm Laws
If log⁡50=9.5\log50=9.5log50=9.5, find an approximation for the following:

Logarithm Laws

Practice: Logarithm Laws
If log⁡20=3.4872\log20=3.4872log20=3.4872, find an approximation for the following:

Logarithm Laws

Practice: Logarithm Laws
If log⁡5=a\log5=alog5=a and log⁡7=b\log7=blog7=b, write each logarithm in terms of aaa and bbb.

Logarithm Laws

Practice: Logarithm Laws
If log⁡3=a\log3=alog3=a and log⁡5=b\log5=blog5=b, write each logarithm in terms of aaa and bbb.

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)}log3​(956​814273​335​​). 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)}loga​(d23​e2a43b​c5​)?