# Practice Problems on logarithms

I) write the following in the logarithmic format

1. $5^3$ = 125

2. $2^4$ = 16

3. $4^{-2}=\frac{1}{16}$

4. $(\frac{1}{2})^{-3}=8$

5. $\sqrt {81}$ = 9

6. $x^{2a}=y$

II) convert the following logarithms into exponents

1. x= $log_3\ y$

2. $log_x\ 1$ =0

3. ln x = 5

4. $log\ 100=2$

5. $log_8\ 2=\frac{1}{3}$

III) Find the values of x

1. $log_5\ 25$ = x

2. $log_{100}\ 1000$ = x

3. $log_{23}\ 1$ = x

4. $log_3\ \frac{1}{27}$ = x

5. $log_x\ 49$ = 7

6. $log_9\ x$ = -3

7. $log_5\ \frac{1}{5}$ = x

8. $log_8\ x=1- \frac{1}{2}$

9. $log_{12}\ 1$ = x

10. $log_{32}\ 2$ = x

11. $log_4\ \frac{1}{32}$ = x

12. $log_{12}\ \frac{1}{144}$ = x

IV) Write the following expressions in terms of logs of x, y and z

1. log $x^3y$

2. log $xyz$

3. log $\sqrt[3]x y^2 z$

4. log $\frac{x}{z^3}y$

5. log $\sqrt[5]{x^3yz^2}$

6. log $x \sqrt{ \frac{\sqrt{x^3y^2}}{z^4}}$

7. log $(x^3y)^{\frac{1}{2}}$

8. log $\frac{x^3}{y}$

V) Solve the following logarithmic equations

1. ln x =-2

2. log (7x+2) = 2

3. log x + log (x+1) = log 4x

4. 2log x = log 2 + log(3x-4)

5. $log_3\ (x+3) + log_3\ (x+2) - log_3\ 10 = log_3\ x$

6. $log_2\ x + log_2\ (x+3) = 1$

VI) If log 2= x, log 3 =y then express the following in x and y

1. log 24

2. log 225

3. log 150

4. $log_7\ 980$

5. log 343

6. log 12.5

7. log 0.2

8. log 15

9. log 2.5

10. $log_7\ 3.5$

VII) Solve the following equations

1. $3^x$ - 1 =8
2. $2^{2x}-2^x-6$=0
3. $3^{1-x}=5^x$
4. $3^{2x-1}+3{x+2}-18$=0
5. $e^{2x}-2e^{x}-15$=0