Use the properties of logarithms to simplify the expressions...

Use the properties of logarithms to simplify the expressions in Exercises 3 and $4 .$
a. $\ln \sin \theta-\ln \left(\frac{\sin \theta}{5}\right)$
b. $\ln \left(3 x^{2}-9 x\right)+\ln \left(\frac{1}{3 x}\right)$
c. $\frac{1}{2} \ln \left(4 t^{4}\right)-\ln 2$

Key Concepts

Algebraic Simplification

Often, the expressions inside the logarithms require algebraic manipulation, such as factoring or reducing fractions, to reveal common terms or facilitate further simplification once the logarithmic properties are applied.

Properties of Logarithms

This concept encompasses the set of rules that govern the operations of logarithms, allowing us to combine or break apart logarithmic expressions. It includes the product rule, quotient rule, and power rule, which are used to simplify expressions by converting sums, differences, or multiples of logarithms into single logarithms.

Logarithm Quotient Rule

The quotient rule states that the difference between two logarithms, ln(a) - ln(b), can be simplified to a single logarithm as ln(a/b). This rule is crucial when simplifying expressions that involve subtraction of logarithms.

Logarithm Product Rule

The product rule allows the sum of two logarithms, ln(a) + ln(b), to be rewritten as a single logarithm, ln(ab). This is useful when combining terms that are added together.

Logarithm Power Rule

According to the power rule, a coefficient in front of a logarithm, k * ln(a), can be brought inside the logarithm as an exponent, resulting in ln(a^k). This rule simplifies expressions where logarithms are multiplied by constants.

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