What are the Properties of Logarithms

What are the Properties of Logarithms

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Properties of Logarithms

Product: \(\log_b(MN)=\log_bM+\log_bN\). Quotient: \(\log_b\left(\frac{M}{N}\right)=\log_bM-\log_bN\). Power: \(\log_b(M^r)=r\log_bM\).

Expanding and Condensing

Expanding breaks one logarithm into several. Condensing combines several logarithms into one.

Domain Checks

When solving logarithmic equations, check that every original log argument is positive.

Properties of Logarithms

Think of this lesson as more than a rule to memorize. Properties of Logarithms is about rewriting exponent questions and solving growth equations. A strong student does not rush to the first formula on the page; they pause, identify the structure of the problem, and then choose the tool that matches that structure. That pause is what prevents most mistakes.

A logarithm answers an exponent question: \(\log_b a=c\) means \(b^c=a\). This translation is the safest first step.

Here is the teacher way to approach the topic. First, read the problem slowly and underline the information that is actually given. Next, name the target: are you finding a value, simplifying an expression, comparing two quantities, solving for a variable, or interpreting a graph? Once the target is clear, the calculation becomes much less mysterious because every step has a job.

  • Rewrite logarithmic statements as exponential statements.
  • Use log properties only when the expression is inside a valid log.
  • Solve the resulting equation.
  • Check domain restrictions.

A helpful habit is to say what each number represents before using it. For example, if a number is a denominator, a radius, a slope, a common difference, or a coefficient, it should not be treated like an ordinary loose number. Its role tells you where it belongs in the formula. This is especially important on ACT-style questions because many wrong answer choices come from using the right numbers in the wrong places.

Another good habit is to keep the work organized vertically. Write one clean line for substitution, one line for simplifying, and one line for the final answer. If the problem has units, keep the units attached. If the problem has signs, exponents, or parentheses, copy them carefully from one line to the next. Most errors in this topic are not caused by a hard idea; they are caused by dropping a negative sign, combining unlike terms, using the wrong denominator, or skipping a check.

When you finish, ask a quick reasonableness question. Should the answer be positive or negative? Should it be larger or smaller than the original number? Does it fit the graph, table, shape, or equation? Can you plug it back into the original problem? This final check turns the lesson from a procedure into understanding.

On a test, the goal is not to write the longest solution. The goal is to write enough clear work that you can see the structure, avoid traps, and recover quickly if one line goes wrong. Practice the examples below with that mindset: identify the type of problem, choose the matching rule, show the substitution, simplify carefully, and check the answer before moving on.

Free printable Worksheets

Exercises for Properties of Logarithms

1) \(Use log properties on \log_2(5x).\)

2) \(Use log properties on \log_3\left(\frac{x}{7}\right).\)

3) \(Use log properties on \log_5(x^4).\)

4) \(Use log properties on \log_2 x+\log_2 3.\)

5) \(Use log properties on \log_4 x-\log_4 y.\)

6) \(Use log properties on 3\log_7 x.\)

7) \(Use log properties on \log_2(8x^3).\)

8) \(Use log properties on \log_5\left(\frac{25x^2}{y}\right).\)

9) \(Use log properties on \log_3 4+2\log_3 x.\)

10) \(Use log properties on \log_6x+\log_6y-\log_62.\)

11) \(Use log properties on \ln\left(\frac{x^2\sqrt y}{e^3}\right).\)

12) \(Use log properties on 2\ln x-\frac12\ln y+4.\)

13) \(Use log properties on \log_2x+\log_24=5.\)

14) \(Use log properties on \log_3x-\log_32=2.\)

15) \(Use log properties on 2\log_5x=4.\)

16) \(Use log properties on \log_4(x-1)+\log_4(x+1)=2.\)

17) \(Use log properties on \log_2(x+6)-\log_2x=2.\)

18) \(Use log properties on \log_3\left(\frac{a^2b^5}{c^4}\right).\)

19) \(Use log properties on \frac13\log_2x+\log_2y-5\log_2z.\)

20) \(Use log properties on \log_3x+\log_3(x-2)=1.\)

 

1)\(Use log properties on \log_2(5x).\)

Step 1: Apply the product, quotient, or power property.

Step 2: expand product.

Answer: \(\log_2 5+\log_2 x\)

2)\(Use log properties on \log_3\left(\frac{x}{7}\right).\)

Step 1: Apply the product, quotient, or power property.

Step 2: expand quotient.

Answer: \(\log_3 x-\log_3 7\)

3)\(Use log properties on \log_5(x^4).\)

Step 1: Apply the product, quotient, or power property.

Step 2: power rule.

Answer: \(4\log_5 x\)

4)\(Use log properties on \log_2 x+\log_2 3.\)

Step 1: Apply the product, quotient, or power property.

Step 2: condense product.

Answer: \(\log_2(3x)\)

5)\(Use log properties on \log_4 x-\log_4 y.\)

Step 1: Apply the product, quotient, or power property.

Step 2: condense quotient.

Answer: \(\log_4\left(\frac{x}{y}\right)\)

6)\(Use log properties on 3\log_7 x.\)

Step 1: Apply the product, quotient, or power property.

Step 2: power rule backward.

Answer: \(\log_7(x^3)\)

7)\(Use log properties on \log_2(8x^3).\)

Step 1: Apply the product, quotient, or power property.

Step 2: expand and evaluate log_2 8.

Answer: \(3+3\log_2x\)

8)\(Use log properties on \log_5\left(\frac{25x^2}{y}\right).\)

Step 1: Apply the product, quotient, or power property.

Step 2: expand product and quotient.

Answer: \(2+2\log_5x-\log_5y\)

9)\(Use log properties on \log_3 4+2\log_3 x.\)

Step 1: Apply the product, quotient, or power property.

Step 2: move 2 as exponent.

Answer: \(\log_3(4x^2)\)

10)\(Use log properties on \log_6x+\log_6y-\log_62.\)

Step 1: Apply the product, quotient, or power property.

Step 2: combine product over quotient.

Answer: \(\log_6\left(\frac{xy}{2}\right)\)

11)\(Use log properties on \ln\left(\frac{x^2\sqrt y}{e^3}\right).\)

Step 1: Apply the product, quotient, or power property.

Step 2: expand all factors.

Answer: \(2\ln x+\frac12\ln y-3\)

12)\(Use log properties on 2\ln x-\frac12\ln y+4.\)

Step 1: Apply the product, quotient, or power property.

Step 2: write 4 as ln e^4.

Answer: \(\ln\left(\frac{e^4x^2}{\sqrt y}\right)\)

13)\(Use log properties on \log_2x+\log_24=5.\)

Step 1: Apply the product, quotient, or power property.

Step 2: condense to log_2(4x)=5.

Answer: \(x=8\)

14)\(Use log properties on \log_3x-\log_32=2.\)

Step 1: Apply the product, quotient, or power property.

Step 2: condense to log_3(x/2)=2.

Answer: \(x=18\)

15)\(Use log properties on 2\log_5x=4.\)

Step 1: Apply the product, quotient, or power property.

Step 2: divide by 2, then rewrite.

Answer: \(x=25\)

16)\(Use log properties on \log_4(x-1)+\log_4(x+1)=2.\)

Step 1: Apply the product, quotient, or power property.

Step 2: condense to x^2-1=16 and check domain.

Answer: \(x=\\sqrt{17}\)

17)\(Use log properties on \log_2(x+6)-\log_2x=2.\)

Step 1: Apply the product, quotient, or power property.

Step 2: condense to (x+6)/x=4.

Answer: \(x=2\)

18)\(Use log properties on \log_3\left(\frac{a^2b^5}{c^4}\right).\)

Step 1: Apply the product, quotient, or power property.

Step 2: expand powers.

Answer: \(2\log_3a+5\log_3b-4\log_3c\)

19)\(Use log properties on \frac13\log_2x+\log_2y-5\log_2z.\)

Step 1: Apply the product, quotient, or power property.

Step 2: condense with exponents.

Answer: \(\log_2\left(\frac{y\sqrt[3]{x}}{z^5}\right)\)

20)\(Use log properties on \log_3x+\log_3(x-2)=1.\)

Step 1: Apply the product, quotient, or power property.

Step 2: condense to x(x-2)=3 and check domain.

Answer: \(x=3\)

Properties of Logarithms Practice Quiz