Solve for $x$. $5^x=23$ A. $x = \square$ (Do not round until the final answer. Then round to four decimal places as needed. Use a comma to separate answers as needed.) B. There is no solution.
Answer (2)
The solution for x in the equation 5 x = 23 is approximately 1.9482 after taking the natural logarithm of both sides, applying logarithmic properties, isolating x , and calculating the final result.
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Take the natural logarithm of both sides of the equation: ln ( 5 x ) = ln ( 23 ) .
Use the logarithm property to rewrite the equation: x ln ( 5 ) = ln ( 23 ) .
Isolate x by dividing both sides by ln ( 5 ) : x = l n ( 5 ) l n ( 23 ) .
Calculate the value of x and round to four decimal places: x ≈ 1.9482 .
1.9482
Explanation
Understanding the Problem We are given the equation 5 x = 23 and asked to solve for x . This is an exponential equation, and we can solve it by taking the logarithm of both sides.
Applying Logarithms Take the logarithm of both sides of the equation. We can use any base for the logarithm, but using the natural logarithm (base e ) or the common logarithm (base 10) is convenient because most calculators have these logarithms built-in. Let's use the natural logarithm: ln ( 5 x ) = ln ( 23 )
Using Logarithm Properties Using the property of logarithms that ln ( a b ) = b ln ( a ) , we can rewrite the equation as: x ln ( 5 ) = ln ( 23 )
Isolating x Now, we isolate x by dividing both sides of the equation by ln ( 5 ) :
x = ln ( 5 ) ln ( 23 )
Calculating the Value of x Using a calculator, we find that: x ≈ 1.609438 3.135494 ≈ 1.9481920934663797 Rounding to four decimal places, we get: x ≈ 1.9482
Final Answer Therefore, the solution for x is approximately 1.9482 .
Examples
Exponential equations like 5 x = 23 are useful in modeling various real-world phenomena, such as population growth, radioactive decay, and compound interest. For example, if you invest money in an account that compounds annually at a certain interest rate, you can use an exponential equation to determine how long it will take for your investment to reach a specific value. Suppose you invest 1000 inana cco u n tt ha tp a ys 5 x ye a rs i s g i v e nb y A = 1000(1.05)^x$. If you want to know how many years it will take to double your investment, you would solve the equation 2000 = 1000 ( 1.05 ) x , which simplifies to 2 = ( 1.05 ) x . Solving for x gives x = l n ( 1.05 ) l n ( 2 ) ≈ 14.21 years.
