Question

Solve the equation $\log 2+2 \log x=\log (5 x+3)$

   Solve the equation $\log 2+2 \log x=\log (5 x+3)$

Algebra and Trigonometry

Algebra and Trigonometry

StemEZ 1st Edition

Chapter 26, Problem 819

Instant Answer

Solved by Expert Ankit Gupta

07/05/2020

Step 1

We can use the properties of logarithms to rewrite the equation as follows: $\log 2 + 2 \log x = \log (5x + 3)$ Using the property $n \log a = \log a^n$, we can rewrite the left side of the equation: $\log 2 + \log x^2 = \log (5x + 3)$ Now, using the property Show more…

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Solve the equation $\log 2+2 \log x=\log (5 x+3)$

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Key Concepts

Properties of Logarithms

The properties of logarithms, such as the product rule and the power rule, are essential for combining or breaking apart logarithmic expressions. The product rule states that log(a) + log(b) equals log(ab), while the power rule indicates that multiplying a logarithm by a constant is equivalent to taking the logarithm of the argument raised to that power. These rules help simplify logarithmic equations to a form that can be solved more easily.

Domain Restrictions in Logarithmic Functions

When working with logarithmic equations, it is necessary to remember that the arguments of all logarithm functions must be positive. This means that any potential solutions must be checked to ensure they make the inputs of the logarithms greater than zero, thereby eliminating any extraneous solutions that do not satisfy the domain.

Transforming to Algebraic Equations

After using logarithmic properties to simplify an equation, the resulting expression often takes the form of an algebraic equation. This transformation allows one to solve the equation through standard algebraic methods, with further simplification or manipulation potentially leading to a quadratic equation.

Quadratic Equations

Quadratic equations, often resulting from the transformation of a logarithmic equation, require techniques such as factoring, completing the square, or applying the quadratic formula to determine the solution. Once solved, these solutions must be cross-checked against any domain restrictions inherent in the original logarithmic equation.

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