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) $\ln x=\left(\log _{10} x\right) /\left(\log _{10} e\right)$ (b) $\ln x=\left(\log _{10} e\right) /\left(\log _{10} x\right)$

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) $\ln x=\left(\log _{10} x\right) /\left(\log _{10} e\right)$
(b) $\ln x=\left(\log _{10} e\right) /\left(\log _{10} x\right)$

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Key Concepts

Algebraic Manipulation of Logarithmic Equations

Solving equations involving logarithms often requires combining logarithmic expressions, applying the change-of-base formula, and transforming the equation into an algebraic form. This manipulation can reveal identities (as in part (a) of the problem) or reduce the equation to a solvable form (as in part (b), where it becomes quadratic in ln x).

Change-of-Base Formula

This principle allows conversion between logarithms of different bases. It states that for any positive numbers a, b, and c (with a, b ? 1), log_a c can be expressed as (log_b c)/(log_b a). This is crucial when relating natural logarithms to common logarithms, as seen in the problem.

Properties of Logarithms

Understanding the basic properties and definitions of logarithms, such as ln x being the natural logarithm of x and log?? x being the common logarithm of x, is key. These properties, including the rule that logarithms only accept positive arguments, underpin the manipulation and solution of logarithmic equations.

Domain Considerations

Logarithms are defined only for positive arguments. This concept is essential to correctly determine the set of valid solutions when solving any logarithmic equation, ensuring that all solutions satisfy x > 0.

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