Solve each logarithmic equation in Exercises 49–92. Be sure to reject any value of x that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. log₅ x+log₅(4x−1)=1

Verified step by step guidance

1. Recall the logarithmic property that allows you to combine the sum of logarithms with the same base: $\log 5_{}a+\log 5_{}b=\log 5_{}(a\cdot b)$. Use this to combine the left side of the equation.
2. Rewrite the equation $\log 5_{}x+\log 5_{}\left(4x-\; 1\right)=\; 1$ as $\log 5_{}\left(x(4x-\; 1)\right)=\; 1$.
3. Use the definition of logarithm to rewrite the equation in exponential form: $5_{}1=x(4x-\; 1)$. This means $5\; =\; 4x^2\; -\; x$.
4. Rearrange the equation to standard quadratic form: $4x^2\; -\; x\; -\; 5\; =\; 0$.
5. Solve the quadratic equation using the quadratic formula: $x\; =\; \backslash frac\{-b\; \backslash pm\; \backslash sqrt\{b^2\; -\; 4ac\}\}\{2a\}$ where $a=4$, $b=-1$, and $c=-5$. After finding the solutions, check each one to ensure it is in the domain of the original logarithmic expressions (i.e., arguments inside the logarithms must be positive). Reject any solution that does not satisfy this domain restriction.

Key Concepts

Properties of Logarithms

Understanding the properties of logarithms, such as the product rule log_b(m) + log_b(n) = log_b(mn), is essential for combining and simplifying logarithmic expressions. This allows the equation log5 x + log5(4x−1) = 1 to be rewritten as a single logarithm, facilitating easier solving.

Domain of Logarithmic Functions

The domain of a logarithmic function log_b(x) requires that the argument x be positive. When solving logarithmic equations, it is crucial to check that all solutions satisfy this condition to avoid extraneous or invalid answers.

Solving Exponential Equations

After combining logarithms into one expression, the equation can be rewritten in exponential form to solve for x. This involves using the definition log_b(a) = c implies a = b^c, enabling the conversion from logarithmic to algebraic form for solution.