Exponent properties

In the following, u and v are variables and m and n are constants (often integers).

1. Product property

[latex]u^m\times u^n=u^{m+n}[/latex]

For example, [latex]u^2\times u^3=u^{2+3}=u^5[/latex]

2. Quotient property

[latex]\frac{u^m}{u^n}=u^{m-n}[/latex]

For example, [latex]\frac{u^5}{u^3}=u^{5-3}=u^2[/latex]

3. Power of a power property

[latex](u^m)^n=u^{mn}[/latex]

For example, [latex](u^2)^3=u^{2\times 3}=u^6[/latex]

4. Power of a product property

[latex](uv)^m=u^mv^m[/latex]

For example, [latex](uv)^2=u^2v^2[/latex]

5. Power of a quotient property

[latex]\left(\frac{u}{v}\right)^m=\frac{u^m}{v^m}[/latex]

For example, [latex]\left(\frac{u}{v}\right)^2=\frac{u^2}{v^2}[/latex]

6. Negative exponent property

[latex]u^{-m}=\frac{1}{u^m}[/latex]

For example, [latex]u^{-2}=\frac{1}{u^2}[/latex]

7. Reciprocal exponent property

[latex]u^\frac{1}{n}=\sqrt[n]{u}[/latex]

For example, [latex]u^\frac{1}{3}=\sqrt[3]{u}[/latex]

8. Fractional exponent property

[latex]u^\frac{m}{n}=\sqrt[n]{u^m}[/latex]

For example, [latex]u^\frac{2}{3}=\sqrt[3]{u^2}[/latex]

9. Zero exponent property

[latex]u^0=1[/latex]

Logarithm properties

1. Exponential and logarithm equivalence

The following statements are equivalent:

• [latex]v=b^u[/latex]
• [latex]\log_b(v)=u[/latex]

2. Inverse property

• [latex]\log_b(b^u)=u[/latex]
• [latex]b^{\log_b(u)}=u[/latex]

3. Exponent property

[latex]\log_b(u^m)=m\log_b(u)[/latex]

For example, [latex]\log_b(u^2)=2\log_b(u)[/latex]

4. Sum property

[latex]\log_b(u)+\log_b(v)=\log_b(uv)[/latex]

5. Difference property

[latex]\log_b(u)-\log_b(v)=\log_b\left(\frac{u}{v}\right)[/latex]

6. Log of one property

[latex]\log_b(1)=0[/latex]

7. Log of the base property

[latex]\log_b(b)=1[/latex]