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Product, Quotient, and Power Properties of Logarithms. In this section, three very important properties of the logarithm are developed. These properties will allow us to expand our ability to solve many more equations. We begin by assigning \(u\) and \(v\) to the following logarithms and then write them in exponential form:
The logarithmic properties are applicable for a log with any base. i.e., they are applicable for log, ln, (or) for logₐ. The 3 important properties of logarithms are: log mn = log m + log n. log (m/n) = log m - log n. log m n = n log m. log 1 = 0 irrespective of the base.
The properties of logarithms will help to simplify the problems based on logarithm functions. Learn the logarithmic properties such as product property, quotient property, and so on along with examples here at BYJU’S.
Learning Objectives. In this section, you will: Use the product rule for logarithms. Use the quotient rule for logarithms. Use the power rule for logarithms. Expand logarithmic expressions. Condense logarithmic expressions. Use the change-of-base formula for logarithms.
We summarize the Properties of Logarithms here for easy reference. While the natural logarithms are a special case of these properties, it is often helpful to also show the natural logarithm version of each property.
Product Property of Logarithms. A logarithm of a product is the sum of the logarithms: loga(MN) = logaM + logaN. where a is the base, a> 0 and a ≠ 1, and M, N> 0.
14 Μαρ 2024 · Use the Properties of Logarithms to expand the following logarithms. Simplify, if possible. \(\log _{4}\left(2 x^{3} y^{2}\right)\) \( log_2(8x^4) \) Solution: a. \( \begin{array}{rll} \log _{4}\left(2 x^{3} y^{2}\right) &= \log _{4}(2) +\log _{4}(x^3) + \log _{4}(y^2) & \text{Product Property, \(\log _{b} M \cdot N=\log _{b} M+\log _{b} N\).
