Solve the following equation. 25x ra=14 x=E (Type an exact…

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Anu Bhardwaj · Accepted Answer

The equation to solve is  2^{ 5 x + 3} = 14 . The solution for x is { l n ( 14 ) − 3}  ​/ { 5} . Step-by-Step Solution To solve this exponential equation, we need to isolate the variable x . Since the variable is in the exponent, we'll use logarithms. Start with the original equation: 2^{ 5 x + 3} = 14 Take the natural logarithm (ln) of both sides. This is a common and effective method for solving exponential equations. ln ( 2^{ 5 x + 3} ) = ln ( 14 ) Use the logarithm property ln ( a^ b ) = b ⋅ ln ( a ) to bring the exponent down to the front. ( 5 x + 3 ) ln ( 2 ) = ln ( 14 ) Distribute ln ( 2 ) to the terms inside the parentheses. 5 x ln ( 2 ) + 3 ln ( 2 ) = ln ( 14 ) Now, we need to get the term with x by itself. Subtract 3 ln ( 2 ) from both sides of the equation. 5 x ln ( 2 ) = ln ( 14 ) − 3 ln ( 2 ) Finally, divide both sides by 5 ln ( 2 ) to isolate x . x = l n ( 14 ) − 3 l n ( 2 ) /  5 l n ( 2 ) ​ This is a correct "exact answer," but it can be simplified further using another logarithm property  ln ( a ) − ln ( b ) = ln ( a/b ​ ) . x = l n ( 14 ) − l n ( 2^ 3 ) /  5 l n ( 2 ) x = l n ( 14 ) − l n ( 8 ) /  5 l n ( 2 ) ​ x =   l n ( 14 ​/8 ) /  5 l n ( 2 ) x = ​ l n (7/4 )/ 5 l n ( 2 ) This is also an exact answer. The most common and simple exact answer, however, is the one derived in step 6 before any further simplification.​

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Anu Bhardwaj

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