Question

Use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is 1. Where possible, evaluate logarithmic expressions without using a calculator. ﻿12(log⁡5x+log⁡5y)−2log⁡5(x+1)\frac{1}{2} \left( \$$log_5 x$$ + \$$log_5 y$$ \right) - 2 \$$log_5 (x + 1)
21$$​(log5​x+log5​y)−2log5​(x+1)

Accepted Answer

Start by applying the distributive property to the expression: multiply $$ \frac{1}{2}  by $$each term inside the parentheses. This gives $$ \frac{1}{2} \log_5 x + \frac{1}{2} \log_5 y - 2 \log_5 (x + 1) . $$Use the power rule of logarithms, which states that $$ a \log_b M = \log_b (M^a) $$, to rewrite each term with coefficients as exponents inside the logarithms. So, $$ \frac{1}{2} \log_5 x = \log_5 (x^{\frac{1}{2}}) $$, $$ \frac{1}{2} \log_5 y = \log_5 (y^{\frac{1}{2}}) $$, and $$ -2 \log_5 (x + 1) = \log_5 ((x + 1)^{-2}) . $$Now, rewrite the expression as the sum and difference of logarithms: $$ \log_5 (x^{\frac{1}{2}}) + \log_5 (y^{\frac{1}{2}}) + \log_5 ((x + 1)^{-2}) . $$Apply the product rule of logarithms, which states $$ \log_b A + \log_b B = \log_b (AB) $$, to combine the first two terms: $$ \log_5 (x^{\frac{1}{2}} y^{\frac{1}{2}}) + \log_5 ((x + 1)^{-2}) . $$Finally, use the product rule again to combine all terms into a single logarithm: $$ \log_5 \left( x^{\frac{1}{2}} y^{\frac{1}{2}} (x + 1)^{-2} \right) . $$This is the condensed form with coefficient 1.

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Key Concepts

Properties of Logarithms

Condensing Logarithmic Expressions

Evaluating Logarithms Without a Calculator