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Chapter 3: Problem 44

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. $$\ln x+\ln 3$$

Short Answer

Expert verified
The condensation of the expression \(ln x + ln 3\) using the properties of logarithms results in \(ln(3x)\).

Step by step solution

01

Identify the rule

Look at the given expression, it has two logarithms being added. The rule for adding logarithms states that the log of the product of two quantities is the sum of their individual logs: that is, \(ln (a) + ln (b) = ln (a*b)\). The logs are with the same base, base e in this case, so this rule can be applied.
02

Apply the addition logarithm rule

Apply the rule \(ln a + ln b = ln (a*b)\) to the expression. The two terms \(x\) and 3 are added together to condense the expression. Replace \(a\) with \(x\) and \(b\) with 3: \(ln x + ln 3 = ln (x*3)\)
03

Simplify the expression

Simplify the expression inside the log to get \(ln(3x)\), which is the expression written as a single logarithm whose coefficient is 1.

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