Solve for x.
$4^x=21$
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A. The solution is $x=$ $\square$ .
(Type an integer or a decimal. Do not round until the final answer. Then round to four decimal places a
B. The solution is not a real number.
Answer (2)
The solution to the equation 4 x = 21 is approximately x = 2.1962 . This is found by taking the natural logarithm of both sides, applying logarithm properties, isolating x , and calculating the values. Thus, the chosen option is A.
;
Take the logarithm of both sides: ln ( 4 x ) = ln ( 21 ) .
Apply the logarithm property: x ln ( 4 ) = ln ( 21 ) .
Isolate x : x = l n ( 4 ) l n ( 21 ) .
Calculate the approximate value: x ≈ 2.1962 .
Explanation
Problem Analysis We are given the equation 4 x = 21 and asked to solve for x .
Taking the Logarithm To solve for x , we can take the logarithm of both sides of the equation. Using the natural logarithm (ln), we have ln ( 4 x ) = ln ( 21 ) .
Applying Logarithm Properties Using the property of logarithms that lo g ( a b ) = b lo g ( a ) , we can rewrite the equation as x ln ( 4 ) = ln ( 21 ) .
Isolating x Now, we can isolate x by dividing both sides by ln ( 4 ) : x = ln ( 4 ) ln ( 21 )
Calculating the Value of x Using a calculator, we find the decimal approximation of x to four decimal places: x ≈ 2.1962
Examples
Exponential equations like this one are used to model various real-world phenomena, such as population growth, radioactive decay, and compound interest. For example, if a population of bacteria doubles every hour, we can use an exponential equation to predict the population size after a certain number of hours. Similarly, in finance, we can use exponential equations to calculate the future value of an investment with compound interest.
