Angie is working on solving the exponential equation [tex]$23^x=6$[/tex]; however, she is not quite sure where to start. Solve the equation and use complete sentences to describe the steps to solve.
Hint: Use the change of base formula: [tex]$\log _0 y=\frac{\log y}{\log b}$[/tex].
Answer (2)
Take the natural logarithm of both sides of the equation: ln ( 2 3 x ) = ln ( 6 ) .
Apply the power rule of logarithms: x ln ( 23 ) = ln ( 6 ) .
Solve for x : x = l n ( 23 ) l n ( 6 ) .
Approximate the value of x : x ≈ 0.5714 . The solution to the equation 2 3 x = 6 is approximately 0.5714 .
Explanation
Understanding the Problem The problem is to solve the exponential equation 2 3 x = 6 for x . We will use logarithms to solve for x . The hint suggests using the change of base formula, but it's not strictly necessary since we can directly take the logarithm of both sides using any base.
Applying Logarithms Take the natural logarithm (base e ) of both sides of the equation: ln ( 2 3 x ) = ln ( 6 ) .
Using the Power Rule Use the power rule of logarithms, which states that ln ( a b ) = b ln ( a ) . Applying this rule, we get: x ln ( 23 ) = ln ( 6 ) .
Isolating x Now, solve for x by dividing both sides by ln ( 23 ) : x = ln ( 23 ) ln ( 6 ) .
Calculating the Result To find an approximate value for x , we can use a calculator to evaluate the logarithms: ln ( 6 ) ≈ 1.79176 ln ( 23 ) ≈ 3.13549 Therefore, x ≈ 3.13549 1.79176 ≈ 0.5714 .
Final Answer The solution to the equation 2 3 x = 6 is x = l n ( 23 ) l n ( 6 ) , which is approximately 0.5714.
Examples
Exponential equations are used in various real-world applications, 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 such equations helps you determine how long it will take for your investment to reach a certain value. Understanding exponential equations is also crucial in fields like biology (population growth), physics (radioactive decay), and finance (compound interest).
To solve the equation 2 3 x = 6 , take the natural logarithm of both sides, apply the power rule, and isolate x . This leads to the solution x = l n ( 23 ) l n ( 6 ) ≈ 0.5714 .
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