- Use the basic properties of logarithms to simplify expressions and solve equations
- Combine or separate logarithms using the product and quotient rules
- Use the power rule to simplify logarithms with exponents
- Break down or combine complicated logarithm expressions into simpler forms
- Use the change-of-base formula to calculate and simplify logarithms with different bases
Basic Properties of Logarithms
Mathematical properties and rules are essential tools for manipulating equations and expressions. While you may be familiar with properties of real numbers and exponents, logarithms have their own set of important properties. These logarithmic properties are fundamental for simplifying, expanding, and solving logarithmic expressions and equations. The zero property, identity property, and inverse property of logarithms form the foundation for understanding more complex logarithmic operations.
[latex]\log_b\left(x\right)=y\ \Leftrightarrow \ {b}^{y}=x,\ \text{}b>0,\ b\ne 1[/latex]
That is, to say that the logarithm to base [latex]b[/latex] of [latex]x[/latex] is [latex]y[/latex] is equivalent to saying that [latex]y[/latex] is the exponent on the base [latex]b[/latex] that produces [latex]x[/latex].
Note that the base [latex]b[/latex] is always a positive number other than [latex]1[/latex] and that the logarithmic and exponential forms “undo” each other.
zero property of logarithms
[latex]\log_b(1)=0[/latex]
This means that the logarithm of [latex]1[/latex] to any base [latex]b[/latex] (where [latex]b \gt 0[/latex] and [latex]b \ne 1[/latex]) is always [latex]0[/latex].
The Zero Property of Logarithms, [latex]\log_b(1)=0[/latex], holds because we can rewrite it as [latex]b^0 = 1[/latex], showing that any base [latex]b[/latex] raised to the power of [latex]0[/latex] equals [latex]1[/latex].
identity property of logarithms
[latex]\log_b(b)=1[/latex]
The Identity Property of Logarithms, [latex]\log_b(b)=1[/latex], holds because, any base [latex]b[/latex] raised to the power of [latex]1[/latex] equals itself, i.e., [latex]b^1 = b[/latex].
Use the the fact that exponentials and logarithms are inverses to prove the zero and identity exponent rule for the following:
1. [latex]{\mathrm{log}}_{5}1=0[/latex]
2. [latex]{\mathrm{log}}_{5}5=1[/latex]
inverse property of logarithms
The inverse property of logarithms involves both logarithms and exponents, showing how they undo each other.
The properties are:
- [latex]b^{\log_b(x)}=x[/latex]
This means that if you take the logarithm of a number and then use that result as the exponent for the base of the logarithm, you get the original number.
- [latex]\log_b(b^x)=x[/latex]
This means that if you have a base raised to a power and then take the logarithm of that result, you get the exponent.
- [latex]\mathrm{log}\left(100\right)[/latex]
- [latex]{e}^{\mathrm{ln}\left(7\right)}[/latex]