4.3 Properties of Logarithms
What is the change-of-base property used for?
The change-of-base property is used to write a logarithm in terms of quantities that can be evaluated with a calculator. Because calculators contain keys for common (base 10) and natural (base e) logarithms, we will frequently introduce base 10 or base e.
What is the product rule for logarithms?
The logarithm of a product is the sum of the logarithms. Example: log(10x)= log 10 + log x = 1 + log x (because log(b)^b = 1, then log 10 = 1.
How do you Condense Logarithmic Expressions?
To condense logarithmic functions, write the sum or difference of two or more logarithmic expressions as a single logarithmic expression by using the reversed properties of logarithms.
Example of condensing a logarithmic expression when the coefficients of logarithms aren't one.
Coefficients of logarithms must be 1 before you can condense them using the product and quotient rules. For example, to condense 2 ln x + ln(x+1), the coefficient of the first term must be one. Use the power rule to rewrite the coefficient as an exponent.
What is the quotient rule for logarithms?
Example: log (x/2) = log x - log 2.
What is the Change-of-Base Property?
In the change-of-base property, base b is the base of the original logarithm. Base a is a new base that we introduce. Thus, the change-of-base property allows us to change from base b to any new base a, as long as the newly introduced base is a positive number not equal to 1.
When is a logarithmic expression expanded?
When using the product, quotient, or power rule. A logarithmic expression is expanded when the product rule is used to write a single logarithm as the sum of two logarithms. A logarithmic expression is also expanded when the quotient rule is used to write a single logarithm as the difference of two logarithms. Finally a logarithmic expression is expanded when the power rule is used to "pull the exponent to the front." When expanding a logarithmic expression it may be necessary to use more than one property of logarithms. When expanding a logarithmic expression determine whether the rewriting has changed the domain of the expression.
What is the power rule for logarithms?
When we use the power rule to "pull the exponent to the front." we say that we are expanding a logarithmic expression.