Find all the real-number roots of each equation. In each...

Find all the real-number roots of each equation. In each case, give an exact expression for the root and also (where appropriate) a calculator approximation rounded to three decimal places.
(a) $\ln \left(x^{3}\right)=3 \ln x$
(b) $(\ln x)^{3}=3 \ln x$

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Substitution Method

Substitution is a technique used to simplify complex equations by introducing a new variable. For example, setting u = ln x can transform a complicated logarithmic equation into a polynomial equation in u, which is often easier to solve.

Algebraic Factoring

Factoring is a fundamental algebraic process used to break down expressions, especially polynomial equations, into simpler products. This technique, along with applying the zero product property, helps in finding all possible roots of an equation.

Properties of Logarithms

This concept involves the rules that govern logarithmic expressions, such as the power rule which states that ln(a^b) = b ln(a). Such properties allow simplification and manipulation of logarithmic equations, making it easier to isolate the variable of interest.

Domain of Logarithmic Functions

Logarithmic functions are only defined for positive arguments. When solving equations with logarithms, it is essential to ensure that the solutions satisfy the condition that the argument of any logarithm is greater than zero.

Transcript

Please give Ace some feedback