The solution set of (log5 x)^2+log5 x(5 / x)=1 is (a) {1,5} (b) {1,1 / 25} (c) {1,5,1 / 25} (d

step 1 In problem 43 we want to get the solution set of this equation first we should get all log to be

The solution set of (log5 x)^2+log5 x(5 / x)=1 is (a)...

Key Concepts

Solving Logarithmic Equations

Solving equations involving logarithms often requires combining the properties of logarithms with algebraic techniques. This might include applying the change-of-base formula, combining like terms, exponentiating both sides of an equation to eliminate logarithms, and eventually reducing the problem to a polynomial or quadratic equation that can be solved by standard methods.

Logarithmic Functions

Logarithmic functions, defined as the inverse of exponential functions, are a central concept in algebra. Their basic properties, such as the product, quotient, and power rules, allow for the simplification and manipulation of expressions involving logs. Understanding these properties is essential when solving equations that contain logarithms.

Change-of-Base Formula

The change-of-base formula is a key tool for rewriting logarithms in different bases. This formula states that log_b(a) can be rewritten as log_c(a) / log_c(b) for any positive base c different from 1. It is particularly useful when dealing with logarithms of non-standard bases or when the equation contains logarithms with different bases.

Domain of Logarithmic Functions

The domain of a logarithmic function consists of all positive real numbers, since the argument of a logarithm must be positive. When solving logarithmic equations, it is crucial to consider these domain restrictions to avoid including extraneous solutions that do not satisfy the original equation.

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