Exercise 2.2 — Class 9 Mathematics Solutions

Solution

1. State the definition of a logarithm.

   $$a^y=x⟺{\log }_{a}x=y$$

2. Identify the base, exponent, and value.

   • Base: $a=10$
   • Exponent: $y=3$
   • Value: $x=1000$

3. Write in logarithmic form.

   $${\log }_{10}1000=3$$

Solution

1. State the definition of a logarithm.

   $$a^y=x⟺{\log }_{a}x=y$$

2. Identify the components.

   • Base $a=2$
   • Exponent $y=8$
   • Value $x=256$

3. Convert to logarithmic form.

   $${\log }_{2}256=8$$

Solution

1. State the conversion rule.

   $$a^y=x⟺{\log }_{a}x=y$$

2. Identify the components.

   • Base $a=3$
   • Exponent $y=-3$
   • Value $x=\frac{1}{27}$

3. Write in logarithmic form.

   $${\log }_3\left(\frac{1}{27}\right)=-3$$

Solution

1. State the conversion rule.

   $$a^y=x⟺{\log }_{a}x=y$$

2. Identify the components.

   • Base $a=20$
   • Exponent $y=2$
   • Value $x=400$

3. Write in logarithmic form.

   $${\log }_{20}400=2$$

Solution

1. State the conversion rule.

   $$a^y=x⟺{\log }_{a}x=y$$

2. Identify the components.

   • Base $a=16$
   • Exponent $y=-\frac{1}{4}$
   • Value $x=\frac{1}{2}$

3. Write in logarithmic form.

   $${\log }_{16}\left(\frac{1}{2}\right)=-\frac{1}{4}$$

Solution

1. State the conversion rule.

   $$a^y=x⟺{\log }_{a}x=y$$

2. Identify components.

   • Base $a=11$
   • Exponent $y=2$
   • Value $x=121$

3. Write in logarithmic form.

   $${\log }_{11}121=2$$

Solution

1. State the conversion rule.

   $$a^y=x⟺{\log }_{a}x=y$$

2. Identify components.

   • Exponential equation: $q^r=p$
   • Base $a=q$
   • Exponent $y=r$
   • Value $x=p$

3. Write in logarithmic form.

   $${\log }_{q}p=r$$

Solution

1. State the conversion rule.

   $$a^y=x⟺{\log }_{a}x=y$$

2. Identify components.

   • Base $a=32$
   • Exponent $y=-\frac{1}{5}$
   • Value $x=\frac{1}{2}$

3. Write in logarithmic form.

   $${\log }_{32}\left(\frac{1}{2}\right)=-\frac{1}{5}$$

Solution

1. State the inverse definition.

   $${\log }_{a}x=y⟺a^y=x$$

2. Identify base, exponent, and value.

   • Base $a=5$
   • Exponent $y=3$
   • Value $x=125$

3. Write in exponential form.

   $$5^3=125$$

Solution

1. State the conversion rule.

   $${\log }_{a}x=y⟺a^y=x$$

2. Identify components.

   • Base $a=2$
   • Exponent $y=4$
   • Value $x=16$

3. Write in exponential form.

   $$2^4=16$$

Solution

1. State the conversion rule.

   $${\log }_{a}x=y⟺a^y=x$$

2. Identify components.

   • Base $a=23$
   • Exponent $y=0$
   • Value $x=1$

3. Write in exponential form.

   $${23}^0=1$$

Solution

1. State the conversion rule.

   $${\log }_{a}x=y⟺a^y=x$$

2. Identify components.

   • Base $a=5$
   • Exponent $y=1$
   • Value $x=5$

3. Write in exponential form.

   $$5^1=5$$

Solution

1. State the conversion rule.

   $${\log }_{a}x=y⟺a^y=x$$

2. Identify components.

   • Base $a=2$
   • Exponent $y=-3$
   • Value $x=\frac{1}{8}$

3. Write in exponential form.

   $$2^{-3}=\frac{1}{8}$$

Solution

1. Rewrite with logarithm on left-hand side.

   $${\log }_{9}3=\frac{1}{2}$$

2. State the conversion rule.

   $${\log }_{a}x=y⟺a^y=x$$

3. Write in exponential form.

   $$9^{\frac{1}{2}}=3$$

Solution

1. Understand the definition of a logarithm.

   By the fundamental definition of a logarithm:

   $${\log }_a(x)=y⟺a^y=x$$

   where:

   • $a$ is the base of the logarithm ($a>0,a\ne 1$)
   • $y$ is the exponent / power
   • $x$ is the number / argument ($x>0$)

2. Identify the base, exponent, and argument from the given equation.

   Given equation:

   $$5={\log }_{10}(100000)\text{or}{\log }_{10}(100000)=5$$

   Comparing with ${\log }_a(x)=y$:

   • Base ($a$): $10$
   • Exponent ($y$): $5$
   • Argument ($x$): $100000$

3. Write in exponential form ($a^y=x$).

   Substituting the identified values into $a^y=x$:

   $${10}^5=100000$$

   (Verification: ${10}^5=10\times 10\times 10\times 10\times 10=100000$.)

Solution

1. State the conversion rule.

   $${\log }_{a}x=y⟺a^y=x$$

2. Identify components.

   • Base $a=4$
   • Exponent $y=-2$
   • Value $x=\frac{1}{16}$

3. Write in exponential form.

   $$4^{-2}=\frac{1}{16}$$

Solution

1. Convert to exponential form.

   $$\begin{array}{cc}{\log }_{x}64& =3\\ x^3& =64\end{array}$$

2. Express $64$ as a cube power.

   $$\begin{array}{cc}{x}^3& =4^3\end{array}$$

3. Equate the bases since powers are equal.

   $$\begin{array}{cc}x& =4\end{array}$$

Solution

1. Convert to exponential form.

   $$\begin{array}{cc}{\log }_{5}1& =x\\ 5^x& =1\end{array}$$

2. Express $1$ as a power of base $5$.

   $$\begin{array}{cc}{5}^x& =5^0\end{array}$$

3. Equate exponents.

   $$\begin{array}{cc}x& =0\end{array}$$

Solution

Solution

1. Convert to exponential form.

   $$\begin{array}{cc}{\log }_{10}x& =-3\\ x& ={10}^{-3}\end{array}$$

2. Simplify negative power.

   $$\begin{array}{cc}x& =\frac{1}{{10}^3}\\ x& =\frac{1}{1000}(\text{or }0.001)\end{array}$$

Solution

1. Convert to exponential form.

   $$\begin{array}{cc}{\log }_{4}x& =\frac{3}{2}\\ x& =4^{\frac{3}{2}}\end{array}$$

2. Simplify the rational exponent.

   $$\begin{array}{cc}x& =(2^2{)}^{\frac{3}{2}}\\ x& =2^{2\times \frac{3}{2}}\\ x& =2^3\\ x& =8\end{array}$$

Solution

1. Convert to exponential form.

   $$\begin{array}{cc}{\log }_{2}1024& =x\\ 2^x& =1024\end{array}$$

2. Factorize $1024$ into base $2$.

   $$\begin{array}{cc}1024& =2^{10}\end{array}$$

3. Equate exponents.

   $$\begin{array}{cc}{2}^x& =2^{10}\\ x& =10\end{array}$$