If log√(5) x+log5^n x+log5^4 x+…upto 7 terms =35, then x is equal to (A) 5 (B) 25 (C) 125 (D) No

step 1 We know log the square root of 5x plus log 5 to the power of 1 3rd x, and etc. and 2 log 5 to th

If log√(5) x+log5^n x+log5^4 x+…upto 7 terms =35, then x is...

Key Concepts

Exponential and Logarithmic Equations

Solving equations that involve logarithms usually requires converting the logarithmic equation into its exponential form. This conversion, combined with the properties of exponents, is key to isolating the variable of interest. It ensures that the solution process is clear and systematic, often involving simplifying the expression to determine the required value of the variable.

Series Summation in Logarithmic Equations

When an equation involves a sum of several logarithmic terms, it is often necessary to view that sum as a series. Recognizing patterns in the bases or exponents can allow one to express the sum in a closed form, particularly if the terms follow an arithmetic progression or other systematic pattern, thereby reducing the problem to solving an algebraic equation.

Properties of Logarithms

Logarithms are the inverses of exponential functions and obey several useful properties, such as converting products into sums, powers into multiplications, and so on. These properties are essential for rewriting and combining logarithmic expressions when solving equations that involve sums of logarithms with different bases.

Change of Base Formula

The change of base formula allows one to rewrite a logarithm in any base in terms of logarithms with another common base. It is given by log_a(b) = log_c(b) / log_c(a). This is crucial when dealing with logarithms that have different bases, as it standardizes them and simplifies the process of adding or comparing logarithmic expressions.

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