 ## What are the Properties of Logarithms

### Logarithmic properties

In math, logarithm problems are solved using the properties of the functions that make up logarithms. In basic math, we learned a lot of properties, like commutative, associative, and distributive, that can be used in algebra. When it comes to logarithmic functions, there are five main things to know:

• Quotient Property
• Power rule
• Change of Base Rule
• Reciprocal rule
• Product Property

### Product Property

Let $$b, \ m, \ n$$ be positive integers and $$b \ ≠ \ 1$$, $$log_b \ (mn \ ) \ = \ log_b \ m \ + \ log_b \ n$$.
So, the $$log$$ of the product of two numbers $$m$$ and $$n$$ with base $$b$$ is the same as adding the $$log$$ of $$m$$ and $$n$$ with the same base $$b$$.

Example:

$$log_2 \ (32 \times 16 \ ) \ = \ log_2 \ 32 \ + \ log_2 \ 16 \ = \ log_2 \ 2^5 \ + \ log_2 \ 2^4 \ = \ 5log_2 \ 2 \ + \ 4log_2 \ 2 \ = \ 5 \ + \ 4 \ = \ 9$$

### Quotient Property

Let $$b, \ m, \ n$$ be positive integers and $$b \ ≠ \ 1$$, $$log_b \ (\frac{m}{n}) \ = \ log_b \ m \ - \ log_b \ n$$.
The difference between $$log \ m$$ and $$log \ n$$ with the same base $$b$$ is equal to the logarithm of a quotient of $$m$$ and $$n$$.

Example:

$$log_3 \ (\frac{81}{729}) \ = \ log_3 \ 81 \ - \ log_3 \ 729 \ = \ log_3 \ 3^4 \ - \ log_3 \ 3^6 \ = \ 4log_3 \ 3 \ - \ 6log_3 \ 3 \ = \ 4 \ - \ 6 \ = \ -2$$

### Power Rule

Let $$b, \ m, \ n$$ be positive integers and $$b \ ≠ \ 1$$, $$log_b \ m^n \ = \ nlog_b \ m$$.

The above property says that the logarithm of a positive number $$m$$ to the power of $$n$$ is equal to the product of $$n$$ and the $$log$$ of $$m$$.

Example:

The most important properties of logarithms are the above three. Here are some other properties and examples of how to use them.

### Change of Base Rule

Let $$a, \ b, \ m$$ be positive integers and $$a \ ≠ \ 1$$, $$b \ ≠ \ 1$$, $$log_a \ m \ = \ \frac{log_b \ m}{log_b \ a}$$.

Example:

$$log_5 \ 25 \ = \ \frac{log_2 \ 25}{log_2 \ 5}$$

### Reciprocal Rule

Let $$m, \ n$$ be positive integers and $$m \ ≠ \ 1$$, $$n \ ≠ \ 1$$, $$log_n \ m \ = \ \frac{1}{log_m \ n}$$.

Example:

$$log_3 \ 12 \ = \ \frac{1}{log_{12} \ 3}$$

### Summary:

• $${a^{\log_a⁡b}}={b}$$
• $$log_a⁡1 = 0$$
• $$log_a⁡a = 1$$
• $$log_a⁡(x .y) = \log_a⁡x+\log_a⁡y$$
• $$log_a\frac{x}{y}=\log_a⁡x-\log_a⁡y$$
• $$\log_a \frac{1}{x}=-\log_a⁡x$$
• $$log_a⁡ x^p= p\log_a⁡x$$
• $$log_{x^k}⁡x =\frac{1}{x}\log_a⁡x,$$  for $$k \neq0$$
• $$log_a⁡x =\log_{a^c}x^c$$
• $$log_a⁡x =\frac{1}{\log_x⁡a}$$

### Exercises for Properties of Logarithms

1) Expand this logarithm: $$log_a \ (x \times y \times z)$$

2) Expand this logarithm: $$log_6 \ (2^5 \times 3^2 \times 7)$$

3) Expand this logarithm: $$log_b \ \frac{x^6 \times y^4}{z^3}$$

4) Expand this logarithm: $$log_7 \ \frac{3^5}{z^6 \times y}$$

5) Expand this logarithm: $$log_a \ \frac{1}{3^2}$$

6) Reduce this phrase to a single logarithm: $$log_b \ q \ + \ log_b \ p$$

7) Reduce this phrase to a single logarithm: $$4 \ log_5 \ 3 \ - \ 6 \ log_5 \ 7$$

8) Reduce this phrase to a single logarithm: $$4 \ log_c \ a \ + \ 3 \ log_c \ b$$

9) Reduce this phrase to a single logarithm: $$5 \ log_a \ x \ - \ 7 \ log_a \ y$$

10) Reduce this phrase to a single logarithm: $$7 \ log_b \ q \ + \ 3 \ log_b \ p$$

1) Expand this logarithm: $$log_a \ (x \times y \times z)$$

$$\color{red}{log_a \ (x \times y \times z) \ = \ log_a \ x \ + \ log_a \ y \ + \ log_a \ z}$$

2) Expand this logarithm: $$log_6 \ (2^5 \times 3^2 \times 7)$$

$$\color{red}{log_6 \ (2^5 \times 3^2 \times 7) \ = \ 5 \ log_6 \ 2 \ + \ 2 \ log_6 \ 3 \ + \ log_6 \ 7}$$

3) Expand this logarithm: $$log_b \ \frac{x^6 \times y^4}{z^3}$$

$$\color{red}{log_b \ \frac{x^6 \times y^4}{z^3} \ = \ 6 \ log_b \ x \ + \ 4 \ log_a \ y \ - \ 3 \ log_a \ z}$$

4) Expand this logarithm: $$log_7 \ \frac{3^5}{z^6 \times y}$$

$$\color{red}{log_7 \ \frac{3^5}{z^6 \times y} \ = \ 5 \ log_7 \ 3 \ - \ 6 \ log_7 \ z \ - \ log_7 \ y}$$

5) Expand this logarithm: $$log_a \ \frac{1}{3^2}$$

$$\color{red}{log_a \ \frac{1}{3^2} \ = \ -log_a \ 3^2 \ = \ -2log_a \ 3}$$

6) Reduce this phrase to a single logarithm: $$log_b \ q \ + \ log_b \ p$$

$$\color{red}{log_b \ q \ + \ log_b \ p \ = \ log_b \ (q \times p)}$$

7) Reduce this phrase to a single logarithm: $$4 \ log_5 \ 3 \ - \ 6 \ log_5 \ 7$$

$$\color{red}{4 \ log_5 \ 3 \ - \ 6 \ log_5 \ 7 \ = \ log_5 \ \frac{3^4}{7^6}}$$

8) Reduce this phrase to a single logarithm: $$4 \ log_c \ a \ + \ 3 \ log_c \ b$$

$$\color{red}{4 \ log_c \ a \ + \ 3 \ log_c \ b \ = \ log_c \ (a^4 \times b^3)}$$

9) Reduce this phrase to a single logarithm: $$5 \ log_a \ x \ - \ 7 \ log_a \ y$$

$$\color{red}{5 \ log_a \ x \ - \ 7 \ log_a \ y \ = \ log_a \ \frac{x^5}{y^7}}$$

10) Reduce this phrase to a single logarithm: $$7 \ log_b \ q \ + \ 3 \ log_b \ p$$

$$\color{red}{7 \ log_b \ q \ + \ 3 \ log_b \ p \ = \ log_b \ (q^7 \times p^3)}$$

## Properties of Logarithms Practice Quiz

### ALEKS Math Full Study Guide

$25.99$12.99

### ISEE Middle Level Math Comprehensive Prep Bundle

$89.99$47.99

### SHSAT Mathematics Formulas

$6.99$6.99

### CHSPE Math Full Study Guide

$25.99$13.99