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Chapter 3: Problem 48

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. $$\log (3 x+7)-\log x$$

Short Answer

Expert verified
The simplified logarithmic expression of \(\log (3x+7) - \log x\) is \(\log \left(\frac{3x+7}{x}\right)\).

Step by step solution

01

Identifying the logarithmic rule to apply

We need to use the quotient rule of logarithms to simplify the given logarithmic expression. This rule states that the difference of two logarithms is equal to the logarithm of the quotient of their arguments. The quotient rule is generally expressed as \(\log_b{A} - \log_b{B} = \log_b{\frac{A}{B}}\).
02

Apply the quotient rule

Applying the quotient rule to \(\log (3x+7) - \log x\) will result in the combination of the two logs into a single log expression: \(\log \left(\frac{3x+7}{x}\right)\)
03

Simplification

The single log expression \(\log\left(\frac{3x+7}{x}\right)\) is the most simplified form of \( \log (3x+7) - \log x \) and it has a coefficient of 1.

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