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) $\left(\log _{10} x\right)^{2}=2 \log _{10} x$ (b) $\log _{10}\left(x^{2}\right)=2 \log _{10} x$

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) $\left(\log _{10} x\right)^{2}=2 \log _{10} x$
(b) $\log _{10}\left(x^{2}\right)=2 \log _{10} x$

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Key Concepts

Domain Restrictions in Logarithmic Equations

Since the logarithm is only defined for positive arguments, it is essential to ensure that any solution to a logarithmic equation results in a positive number when substituted into all logarithms. This concept prevents extraneous solutions and ensures the validity of the results.

Solving Quadratic Equations via Substitution

By letting a variable represent a logarithmic expression (for example, u = log_b(x)), a logarithmic equation can be transformed into a quadratic equation. Solving the quadratic equation using factoring, the quadratic formula, or other methods allows one to find the values of the new variable, which can then be converted back to the original variable.

Logarithmic Functions

This concept covers the understanding that logarithms are the inverses of exponential functions. It includes knowing that for any base b (typically b > 0, b ? 1), the logarithmic function log_b(x) is defined only for x > 0. This understanding is crucial when dealing with any equation that involves logarithms.

Properties of Logarithms

This includes the use of rules such as log_b(x^n) = n log_b(x) and the expansion and simplification of logarithmic expressions. Mastery of these properties allows one to manipulate logarithmic equations into forms that are easier to solve.

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