Chapter 3
Logarithms
In the seventeenth century, a Scottish mathematician John Napier developed method for efficiently performing calculations with larger numbers. He found a method for finding the product of two numbers by adding two corresponding numbers, which is called logarithms.The Richter scale is a logarithmic scale use to measure the magnitude of an earthquate.pH is another logarithmic scale used to describe how acidic or basic an aqueous solution is. (၁၇)ရာစုတွင်စကော့တလန်သင်္ချာပညာရှင် John Napier သည်အလွန်ကြီးသောကိန်းများကိုထိရောက်စွာကိုင်တွယ်နိုင်မည့်နည်းလမ်းကိုတီထွင်ခဲ့သည်။ကိန်းနှစ်လုံးမြှောက်လဒ်ရှာဖွေရန် logarithm ဟုခေါ်သောသက်ဆိုင်ရာကိန်းနှစ်လုံးပေါင်းထည့်သည့်နည်းလမ်းတစ်ခုရှာဖွေတွေ့ရှိခဲ့သည်။ Richter scale သည်မြေငလျင်တစ်ခု၏ပမာဏကိုတိုင်းတာရန်သုံးသော Logarithmic scale တစ်ခုဖြစ်သည်။ pH scale သည် aqueous solution တစ်ခုတွင်အက်ဆစ်ဓာတ်(သို့မဟုတ်)ဘေ့စ်ဓာတ်မည်မျှပါဝင်ကြောင်းဖော်ပြရာတွင်သုံးသည့်အခြားသော logarithmic scale တစ်ခုဖြစ်သည်။ သိပ္ပံပညာရှင်များသည်အလွန်ကြီးသော(သို့မဟုတ်)အလွန်သေးသောကိန်းဂဏန်းများ(ဉပမာအားဖြင့် the speed of light in vacuum is$299,792,458$metres per second and the mass of an electron is $0.000000000000000000000000000000910938$ kilograms)ကိုကိုင်တွယ်ဖြေရှင်းရာတွင် (scientific notation) ဟုခေါ်သောပို၍ကျစ်လစ်သောကိန်းပုံစံကိုတီထွင်အသုံးပြုလာကြသည်။Example 1.
Write the following numbers in scientific notation. (a) $14,753$ (b) $0.00632$ (c) $0.23$ (d) $0.00000912$ (e) $1,000,000$Example 2.
Express the following numbers in ordinary decimal form. (a) $7.354\times10^{5}$ (b) $3.2\times10^{-1}$Example 3.
Evaluate each of the followings and express the results in scientific notation: (a) $4.215\times10^{-2}+3.2\times10^{-4}$ [Addition] (b) $8.97\times10^{4}-2.62\times10^{3}$ [Subtraction] (c) ($6.73\times10^{-5}$)($2.91\times10^{2}$) [Multiplication] (d) $\dfrac{6.4\times10^{6}}{1.92\times10^{2}}$ [Division] (e) ($6.5\times10^{-3})^{2}$ [Power] (f) $\sqrt{3.6\times10^{5}}$ [Root]Example 4.
Evaluate $\dfrac{2,750,000\times0.015}{750}$ by transforming each number to scientific notation.Example 5.
Write each of the following in logarithmic form. (a) $10^{-2}=0.01$ (b) $4^{\frac{1}{2}}=2$ (c) $7^{2}=49$Example 6.
Express each of the following in exponential form. (a) $\log_{2}8=3$ (b) $\log_{10}1=0$ (c) $\log_{5}\left(\dfrac{1}{\sqrt{5}}\right)=-\dfrac{1}{2}$Example 7.
Find the value of each logarithm. (a) $\log_{5}25$ (b) $\log_{2}16\sqrt{2}$ (c) $\log_{\frac{1}{2}}8$Example 8.
Evaluate each expression. (a) $3^{\log _{3} 7}+\log _{5} 125$ (b) $\log _{7} 7^{9}-\log _{3} \dfrac{1}{9}$ (c) $\log _{5}\left(\log _{2}\left(\log _{3} 9\right)\right)$ (d) $10^{2+\log _{10} 5}$Example 9.
(a) Given that $10^{0.3010}=2$, find the value of $\log_{10}16$. (b) Solve $\log_{3}(x^{2}-1)=2$.Basic properties of exponents and logarithms can be summarized as follow. ထပ်ညွှန်း၏အခြေခံဂုဏ်သတ္တိများနှင့် logarithm ၏အခြေခံဂုဏ်သတ္တိများကိုယှဉ်တွဲလေ့လာနိုင်သည်။
| Properties | For exponents | For Logarithms |
| One-to-one Property | If $b^{x}=b^{y}$, then $x=y$ |
If $\log_{b}M=\log_{b}N$, then $M=N$. |
| Product Property | $b^{x}.b^{y}=b^{x +y}$ | $\log_{b}(M N)=\log_{b}M+\log_{b}N$ |
| Quotient Property | $\dfrac{b^{x}}{b^{y}}=b^{x -y}$ | $\log_{b}\dfrac{M}{N}=\log_{b}M-log_{b}N$ |
| Power Property | $(b^{x})^{y}=b^{x y}$ | $\log_{b}N^{p}=p\log_{b}N$ |
Example 10.
If $p=\log_{b}2$, $q=\log_{b}3$ and $r=\log_{b}5$, write $\log_{b}\dfrac{5\sqrt{2}}{2}$ in terms of $p$, $q$ and $r$.Example 11.
Using $\log_{2}3=1.5850$, find the values of (a) $\log_{2}24$ (b) $\log_{2}0.75$Example 12.
Write each expression as a single logarithm. (a) $2+3 \log _{5} x^{2}$ (b) $\log _{3} 2+\log _{9} 81$ (c) $\log _{b}(3 x)+\log _{b}(4 y)-\log _{b}(2 z)$ (d) $-\log _{3}(2 s)+\dfrac{1}{2} \log _{3}\left(4 t v^{3}\right)-2 \log _{3}(5 u)$Example 13.
Solve the following equations for $x$. $\log _{2}\left(3 x^{2}-1\right)-\log _{2}(2 x)=0$Example 14.
Suppose that $\log _{b}\left(x y^{2}\right)=4$ and $\log _{b}\left(\dfrac{x^{3}}{y}\right)=5$. (a) Write the equation connecting $\log _{b} x$ and $\log _{b} y$. (b) Find the values of $\log _{b} x$ and $\log _{b} y$. (c) Find $\log _{b}\left(y^{5} \sqrt{x}\right)$. (d) Write $x$ and $y$ in terms of $b$.Example 15.
Given that $\log _{p} x=20$ and $\log _{p} y=5$, find $\log _{y} x$ and $\log _{x} y$.Example 16.
Find the value of (a) $2^{\frac{\log _{5} 3}{\log _{5} 2}}$ (b) $5^{\frac{1}{\log _{7} 5}}$ (c) $\log _{3} 5 \times \log _{25} 27$Example 17.
Solve the equation $\log _{3} x=3-2 \log _{x^{3}}$, where $x>0$ and $x \neq 1$.Example 18.
Solve the logarithmic equation $\log _{3} x=\log _{9}(x+6)$.Example 19.
Given that $\log _{10} 7=0.8451$; what are the characteristics and the mantissas of $\log _{10} 0.007$ and $\log _{10} 700$ ?Example 20.
Two nonnegative real numbers $A$ and $P$ are related by the formula $A=Pe^{0.085t}$. Given that $\ln 2=0.6931$, find the value of $t$ for which $A$ becomes$200%$ of $P$.Example 21.
Given that $\log _{10} 2=0.3010$, $\log _{10}9.87=0.9943$ and $\log _{10}8.5=0.9294$; evaluate $\dfrac{200 \times98.7 \times 85}{8.5^{3}}$ by using logarithms.Exercise 3.1
- How many significant figures are there in each of the following numbers? (a) $2.175$ (b) $0.2175$ (c) $0.0075$ (d) $89400$ (e) $0.00046$
- Write in scientific notation. (a) $24.86$ (b) $2.486$ (c) $0.2486$ (d) $0.002486$ (e) $0.073$ (f) $0.0086$ (g) $0.934$ (h) $7$ (i) $0.00056857$ (j) $6.843250$
- Write each number in ordinary decimal form. (a) $7.84\times 10^{4}$ (b) $7.89\times 10^{-4}$ (c) $2.25\times 10^{5}$ (d) $4.01\times 10^{-3}$
- Simplify and give the answers in scientific notation. (a) $2.3 \times 10^{2}+1.7 \times 10^{2}$ (b) $4.6 \times 10^{-3}-2.5 \times 10^{-3}$ (c) $\left(4.5 \times 10^{6}\right) \times\left(1.5 \times 10^{-2}\right)$ (d) $\dfrac{7.6 \times 10^{5}}{1.9 \times 10^{-2}}$
- Compute using scientific notation. (a) $\dfrac{2.5 \times 10^{2}}{0.25 \times 0.002}$ (b) $\dfrac{33,000,000 \times 0.4}{1.1 \times 30}$ (c) $\dfrac{50 \times 0.014 \times 0.30}{10500}$ (d) $\dfrac{7000 \times 80 \times 300}{400}$
Exercise 3.2
- Write the following equations in Logarithmic form. (a) $ 3^{4} =81$ (b) $ 9^{\frac{3}{2}}=27$ (c) $10^{-3}=0.001$ (d) $3^{-1}=\dfrac{1}{3}$ (e) $\left(\dfrac{1}{4}\right)^{-3} =64$
- Write the following equations in exponential form. (a) $ \log _{10} 3=0.4771$ (b) $\log _{6} 0.001=-3.855$ (c) $ \log _{44} 12=\dfrac{1}{2}$ (d) $-5=\log _{3} \dfrac{1}{243}$ (e) $\log _{x} 7=y^{2}$, where, $0<x<1$
- Solve the following equations. (a) $ \log _{7} 49=x$ (b) $\log _{x} 10=1$ (c) $ \log _{\sqrt{3}} x=2$ (d) $\log _{\sqrt{3}} x=2$ (e) $x^{\log _{x} x}=5$
- Evaluate. (a) $9^{\log _{9} 2}+3^{\log _{3} 8}$ (b) $\log _{4} 4^{5} \times \log _{10} 10^{2}$ (c) $7^{\log _{7} 9}+\log _{2}\left(\dfrac{1}{2}\right)$ (d) $\log _{\frac{1}{2}}\left(\dfrac{1}{8}\right)-4 \log _{10} 10$ (e) $10^{1-\log _{10} 3}$
- PFind the value of $x$ in each of the following problems. (a) $\log _{3}(2 x-5)=2$, where $x>\dfrac{5}{2}$ (b) $\log _{77}\left(\log _{7} x\right)=0$, where $x>0$ (c) $8+3^{x}=10$, given that $\log_{3}4=1.2619$
Exercise 3.3
- Replace $□$ with the appropriate number. (a) $\log _{3} 24=\log _{3} 6+\log _{3} □$ (b) $\log _{5} 24=\log _{5} 60+\log _{5} □$ (c) $\log _{2} □=3 \log _{2} 3$ (d) $\log _{10} 9=□ \log _{10} 3$ (e) $\log _{8} 5=\log _{8} □-\log _{8} 11$
- Write each expression as a single logarithm. (a) $\log _{b} 20+\log _{b} 57-\log _{b} 114$ (b) $3 \log _{b} 8-\log _{b} 12=\log _{b} 8^{3}-\log _{b} 12$ (c) $\log _{b} x-2 \log _{b} y-\log _{b} a$ (d) $\log _{2} 3+\log _{4} 15$
- Write each expression in terms of $\log_{2}$, $\log_{3}$, $\log_{5}$. (a) $ \log _{b} 8$ (b) $\log _{b} 15$ (c) $\log _{b} 270$ (d) $\log _{b} \dfrac{27 \sqrt[3]{5}}{16}$ (e) $\log _{b} \dfrac{216}{\sqrt[3]{32}}$ (f) $\log _{b}(648 \sqrt{125})$
- Evaluate each expression. (a) $ \log _{7} 49=x$ (b) $\log _{x} 10=1$ (c) $ \log _{\sqrt{3}} x=2$ (d) $\log _{\sqrt{3}} x=2$ (e) $x^{\log _{x} x}=5$ (f) $\log _{\sqrt{3}} x=2$ (g) $x^{\log _{x} x}=5$ (h) $\log _{\sqrt{3}} x=2$
- Use $\log_{2}=0.3010$ and $\log_{3}=0.4771$ to evaluate each of the following expressions. (a) $\log _{10} 6$ (b) $\log _{10} 1.5$ (c) $ \log _{10} \sqrt{3}$ (d) $\log _{10} 4$ (e) $\log _{10} 4.5$ (f) $\log _{10} 8$ (g) $\log _{10} 18$ (h) $\log _{10} 5$
- Solve the following equations for $x$. (a) $\log _{a} \dfrac{18}{5}+\log _{a} \dfrac{10}{3}-\log _{a} \dfrac{6}{7}=\log _{a} x$ (b) $\log _{b} x=2-a+\log _{b}\left(\dfrac{a^{2} b^{a}}{b^{2}}\right)$ (c) $\log x^{3}-\log x^{2} =\log 5 x-\log 4 x$ (d) $\log _{10} x+\log _{10} 3 =\log _{10} 6$ (e) $8 \log x=\log a^{\frac{3}{2}}+\log 2-\dfrac{1}{2} \log a^{3}-\log \dfrac{2}{a^{4}}$
- Given that $\log_{10}5=0.6990$ and $\log_{10}x=0.2330$. What is the value of $x$?
- Show that if $\log_{e}I=-\dfrac{R}{L}t+\log_{e}I_{0}$, then $I=I_{0}e^{-\frac{Rt}{L}}$.
- Show that if $\log_{b}y=\dfrac{1}{2}\log_{b}x+c$, then $y=b^{c}\sqrt{x}$.
- Show that (a) $\dfrac{1}{4} \log _{10} 8+\dfrac{1}{4} \log _{10} 2=\log _{10} 2$ (b) $4 \log _{10} 3-2 \log _{10} 3+1=\log _{10} 90$
- Show that (a) $a^{2 \log _{a} 3}+b^{3 \log _{b} 2}=17$ (b) $3 \log _{6} 1296=2 \log _{4} 4096$
- Given that $\log_{10}12=1.0792$ and $\log_{10}24=1.3802$, deduce the values of $\log_{10}2$ and $\log_{10}6$.
- If $\log_{x}a=5$ and $\log_{3a}3a=9$, find the values of $a$ and $x$.
15.(a) If $\log_{2}\left(4x - 4\right)=2$, find the value of $\log_{4}x$. (b) Prove that if $\dfrac{1}{2}\log_{3}M+3\log_{3}N=1$ then $MN^{6}=9$.
Exercise 3.4
- If $\log_{a}b+\log_{b}a^{2}=3$, find $b$ in terms of $a$.
- Show that (a) $\log_{4}x=2\log_{16}x$ (b) $\log_{b}x=3\log_{b^{3}}x$ (c) $\log_{2}x=(1+\log_{2}3)\log_{6}x$
- If $a=\log_{b}c$, $b=\log_{c}a$ and $c=\log_{a}b$, prove that $a b c =1$.
- Show that (a) $\left(\log _{10} 4-\log _{10} 2\right) \log _{2} 10=1$ (b) $2 \log _{2} 3\left(\log _{9} 2+\log _{9} 4\right)=3$
- Compute (a) $3^{\log _{2} 5}-5^{\log _{2} 3}$ (b) $4^{\log _{2} 3}$ (c) $2^{\log _{2 \sqrt{2}} 27}$
Exercise 3.5
- Given that $\log 2.345=0.3701$. What are the characteristics and the mantissas of each of the followings? (a) $\log 234,500$ (b) $\log 0.0002345$
- Given that $\log_{10}2.74=0.4377$, $\log_{10}2.83=0.4518$, $\log_{10}5.97=0.7760$, $\log_{10}6.21=0.7931$, $\log_{10}8.18=0.9128$ and $\log_{10}9.27=0.9671$, compute (a) $\left(\dfrac{28.3}{597 \times 621}\right)^{2}$ (b) $\dfrac{274^{\frac{1}{3}}}{927 \times 818}$ (c) $\dfrac{28.3 \sqrt{0.621}}{597}$
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