Logarithm: Miscellaneous Exercise


Definition: Logarithm


Let $N$ and $b$ be positive real numbers, with $b \neq 1$. Then the logarithm of $N$ (with respect) to the base $b$ is the exponent by which $b$ must be raised to yield $N$, and is denoted by $\log _{b} N$

Rules of Logarithms


$\begin{array}{ll} \text{L}1. & N=b^{\log _{b} N}\\\\ \text{L}2. & x=\log _{b} b^{x}\\\\ \text{L}3. & \log _{b} b=1\\\\ \text{L}4. & \log _{b} 1=0\\\\ \text{L}5. & \log _{b}(M N)=\log _{b} M+\log _{b} N\\\\ \text{L}6. & \log _{b} N^{p}=p \log _{b} N\\\\ \text{L}7. & \log _{b}\left(\displaystyle\frac{M}{N}\right)=\log _{b} M-\log _{b} N\\\\ \text{L}8. & \log _{a} N=\displaystyle\frac{\log _{b} N}{\log _{b} N}\\\\ \text{L}9. & \log _{a} N=\displaystyle\frac{1}{\log _{N} a}\\\\ \text{L}10. & \log _{a^{p}} N=\displaystyle\frac{1}{p} \log _{a} N\\\\ \text{L}11. & a^{\log _{k} b}=b^{\log _{k} a} \end{array}$

Common Logarithm


The logarithm of $N$ to the base $10\left(\log _{10} N\right)$ is said to be a common logarithm, and is usually written as $\log N$ (omitting the base). where $n$ is called the characteristic and $\log a$ is called the mantissa of $\log N$.

$\begin{array}{|l|} \hline\log _{10} N=\log N\\ \hline \end{array}$


If $\quad N=a \times 10^{n}$,

then $\quad \log N=\log \left(a \times 10^{n}\right)=\log 10^{n}+\log a=n+\log a$ where $n$ is called the characteristic and $\log a$ is called the mantissa of $\log N$.

Note that $n$ is an integer and $1 \leq a<10$.

Euler's Number


As a positive integer $n$ become very large, the value of $\left(1+\displaystyle\frac{1}{n}\right)^{n}$ approaches an irrational number, which is denoted by $e$.

Natural Logarithm


The logarithm of $N$ to the base $e$ is called a natural logarithm, and is denoted by $\ln N$.

$\begin{array}{|l|} \hline\log _{e} N=\ln N\\ \hline \end{array}$

  1. Write the following expressions in terms of $\log x$, $\log y$ and $\log z$.
  2. (a) $\log (x^{2} y)$ Show Solution
    (b) $\log \displaystyle\frac{x^{3} y^{2}}{z}$ Show Solution
    (c) $\log \displaystyle\frac{\sqrt{x} \sqrt[3]{y^{2}}}{z^{4}}$ Show Solution
    (d) $\log (x y z)$ Show Solution
    (e) $\log \left(\displaystyle\frac{x}{y z}\right)$ Show Solution
    (f) $\log \left(\displaystyle\frac{x}{y}\right)^{2}$ Show Solution
    (g) $\log \left(x y\right)^{\frac{1}{3}}$ Show Solution
    (h) $\log (x \sqrt{z})$ Show Solution
    (i) $\log \displaystyle\frac{\sqrt[3]{x}}{\sqrt[3]{y z}}$ Show Solution
    (j) $\log \sqrt[4]{\displaystyle\frac{x^{3} y^{2}}{z^{4}}}$ Show Solution
    (k) $\log \left(x \sqrt{\displaystyle\frac{\sqrt{x}}{z}}\right)$ Show Solution
    (l) $\log \sqrt{\displaystyle\frac{x y^{2}}{z^{8}}}$ Show Solution

  3. Prove the following statements.
  4. (a) $\quad\log _{\sqrt{b}} x=2 \log _{b} x$ Show Solution
    (b) $\quad\log _{\frac{1}{\sqrt{b}}} \sqrt{x}=-\log _{b} x$ Show Solution
    (c) $\quad\log _{b^{4}} x^{2}=\log _{b} \sqrt{x}$ Show Solution

  5. Given that $\log 2=x, \log 3=y$ and $\log 7=z$, express the following expressions in terms of $x, y$, and $z$.
  6. (a) $\log 12$ Show Solution
    (b) $\log 200$ Show Solution
    (c) $\log \displaystyle\frac{14}{3}$ Show Solution
    (d) $\log 0.3$ Show Solution
    (e) $\log 1.5$ Show Solution
    (f) $\log 10.5$ Show Solution
    (g) $\log 15$ Show Solution
    (h) $\log \displaystyle\frac{6000}{7}$ Show Solution

  7. Solve the following logarithmic equations.

  8. (a) $\ln x=-3$ Show Solution
    (b) $\log (3 x-2)=2$ Show Solution
    (c) $2 \log x=\log 2+\log (3 x-4)$ Show Solution
    (d) $\log x+\log (x-1)=\log (4 x)$ Show Solution
    (e) $\log _{3}(x+25)-\log _{3}(x-1)=3$ Show Solution
    (f) $\log _{9}(x-5)+\log _{9}(x+3)=1$ Show Solution
    (g) $\log x+\log (x-3)=1$ Show Solution
    (h) $\log _{2}(x-2)+\log _{2}(x+1)=2$ Show Solution

  9. Find the inverse of each of the following functions.

  10. (a) $\quad f(x)=\log _{2}(x-3)-5$

    Show Solution

    (b) $\quad f(x)=3 \log _{3}(x+3)+1$

    Show Solution

    (c) $\quad f(x)=-2 \log 2(x-1)+2$

    Show Solution

    (d) $\quad f(x)=-\ln (1-2 x)+1$

    Show Solution

    (e) $\quad f(x)=2^{x}-3$

    Show Solution

    (f) $\quad f(x)=2 \cdot 3^{3 x}-1$

    Show Solution

    (g) $\quad f(x)=-5 \cdot e^{-x}+2$

    Show Solution

    (h) $\quad f(x)=1-2 e^{-2 x}$

    Show Solution