Log Rules Practice Problems with Answers

Use the exercise below to practice your skills in applying Log Rules. There are ten (10) problems of various difficulty levels to challenge you. Have fun!

Problem 1: Simplify [latex]{\log _2}16 + {\log _2}32[/latex]

Answer

[latex]\color{red}9[/latex]


Problem 2: Simplify [latex]{\log _3}81 – {\log _3}9[/latex]

Answer

[latex]\color{red}2[/latex]


Problem 3: Simplify [latex]{\log _2}\left( {{1 \over 8}} \right) + {\log _3}\left( {{1 \over 9}} \right)[/latex]

Answer

[latex]\color{red}-5[/latex]


Problem 4: Simplify [latex]{\log _3}729 – 2\log_3 {3^3} + {\log _4}16 + 2{\log _4}2[/latex]

Answer

[latex]\color{red}3[/latex]


Problem 5: Simplify [latex]\large{{1 \over 2}\,{\log _2}\,{4^8} – {2 \over 3}\,{\log _3}\,{27^9}}[/latex]

Answer

[latex]\color{red}-10[/latex]


Problem 6: Use the rules of logarithms to condense the expression below as a single logarithmic expression.

log base 2 of 4 * m^2 - log base 2 of 2 * m^4
Answer

[latex]\large{\color{red}{\log _2}\left( {{2 \over {{m^2}}}} \right)}[/latex]


Problem 7: Use the rules of logarithms to write the expression below as a single logarithm.

log base 3 of (2*x^-1) + log base 3 of (12*x^5) + log base 3 of (2/3*x^-3)
Answer

[latex]\color{red}{\log _3}\left( {16x} \right)[/latex]


Problem 8: Use the rules of logarithms to write the expression below as a single logarithm.

log base 10 of (4*y^3) + log base 10 of (5y^4) - log base 10 (2*y^5)
Answer

[latex]\color{red}\log \left( {10{y^2}} \right)[/latex]


Problem 9: Use the rules of logarithms to expand the expression below.

log base 2 [ 24 * (k+2)^3 ]
Answer

[latex]\color{red}3 + {\log _2}\left( 3 \right) + 3{\log _2}\left( {k + 2} \right)[/latex]


Problem 10: Use the rules of logarithms to expand the expression below.

log base 3 of (243*y^2)/(sqrt(x))
Answer

[latex]\color{red}5 + 2{\log _3}\left( y \right) – {1 \over 2}{\log _3}\left( x \right)[/latex]


You may also be interested in these related math lessons or tutorials:

Logarithm Rules

Expanding Logarithms

Condensing Logarithms

Proofs of Logarithm Properties

Solving Logarithmic Equations