Solve $2\left(5^x\right)-15=9$. Round your answer to three decimal places.
$x=$
Answer (2)
To solve the equation 2 ( 5 x ) − 15 = 9 , we isolate the exponential term and simplify to find x = l n ( 5 ) l n ( 12 ) . This calculation gives approximately 1.544 when rounded to three decimal places. The final answer is therefore x ≈ 1.544 .
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Add 15 to both sides: 2 ( 5 x ) = 24 .
Divide both sides by 2: 5 x = 12 .
Take the natural logarithm of both sides: x ln ( 5 ) = ln ( 12 ) .
Solve for x and round to three decimal places: x = l n ( 5 ) l n ( 12 ) ≈ 1.544 .
Explanation
Problem Analysis We are given the equation 2 ( 5 x ) − 15 = 9 and asked to solve for x , rounding the answer to three decimal places.
Isolating the Exponential Term First, we isolate the term with the exponent. Add 15 to both sides of the equation: 2 ( 5 x ) − 15 + 15 = 9 + 15
2 ( 5 x ) = 24
Simplifying the Equation Next, divide both sides of the equation by 2: 2 2 ( 5 x ) = 2 24
5 x = 12
Taking the Natural Logarithm Now, take the natural logarithm (ln) of both sides of the equation: ln ( 5 x ) = ln ( 12 )
Applying Logarithm Properties Use the property of logarithms that ln ( a b ) = b ln ( a ) to bring the exponent down: x ln ( 5 ) = ln ( 12 )
Solving for x Solve for x by dividing both sides by ln ( 5 ) :
x = ln ( 5 ) ln ( 12 )
Calculating the Value of x Calculate the value of x . The result of this operation is approximately 1.5439593106327716.
Rounding the Answer Finally, round the answer to three decimal places: x ≈ 1.544
Examples
Exponential equations like this one are used in various real-world scenarios, such as modeling population growth, radioactive decay, and compound interest. For example, if you invest money in an account that compounds interest continuously, the amount of money you have after a certain time can be modeled by an exponential equation. Solving for the exponent (in this case, x) allows you to determine how long it will take for your investment to reach a certain value.
