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Solve for [tex]x[/tex]:

\[
\log_{e} 2 \cdot \log_{x} 625 = \log_{10} 16 \cdot \log_{e} 10
\]

Answer :

Final answer:

The logarithmic equation was solved by applying the properties of logarithms, specifically base 'e' logarithm which is a natural logarithm, then resolving the equation with arithmetic operations. The solution to the given equation is 'x' equals 5.

Explanation:

The given algebraic logarithmic equation is log base (e) 2 . log base(x) 625 = log base (10) 16 . log base (e) 10. Using the properties of logarithms, this equation can be simplified and solved for 'x'.

First, we understand that a base 'e' logarithm is equivalent to a natural logarithm ('ln'), therefore, the equation simplifies to ln(2) . log base(x) 625 = log(16) . ln(10).

The right-hand side further simplifies to 2 . ln(10) since log(10^m) equals to m and 16 is equal to 16^2.

As per the logarithmic property, the exponent can be placed at the front. Hence, log base(x) 625 can be rewritten as 4log base(x) 5.

Now, we equate the equation ln(2) . 4log base(x) 5 = 2 . ln(10).

This implies that log base(x) 5 = ln(10)/ln(2), and therefore, 'x' equals e^(ln(10)/ln(2)).

Finally, using properties of natural logarithms and exponents, 'x' can be solved as 10/2 leading to 'x' equals 5.

Learn more about Logarithmic Equation here:

https://brainly.com/question/31451863

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